ࡱ>  quwy`q` bjbjqPqP >::|||||||8888p4lE*b($ohx|L@|||555||555$|||0 -7k8o 0E~Oc<O$00O|Dl45RyE$z|||||| Fundamental rules of counting Multiplication Principle: If an operation can be performed in `m' ways and if another operation can be performed in `n' ways then the no of ways in which both the operation can be done simultaneously is m x n ways. Example: To wear 1 hat and 1 t-shirt from, 3 hats H1, H2, H3 and 4 t-shirts S1, S2, S3, S4 Possible ways H1S1, H1S2, H1S3, H1S4, H2S1, H2S2, H2S3, H2S4, H3S1, H3S2, H3S3, H3S4, Addition Principle: If an operation can be performed in `m' ways and if another operation can be performed in `n' ways then the total no. of ways in which either of the two operations can be performed is `m + n' ways. FACTORIAL The continuous product of first n natural numbers is denoted by the symbol n or n! and is read as factorial n or n-factorial. Thus  n! = n(n-1) (n-2) 3.2.1, n ( 1 0! = 1 and Note: Factorial of a negative integer is not defined. AND : Multiplication, OR : Addition. Note: If nothing is mentioned it is always assumed that repetition is not allowed. PROBLEMS ON FUNDAMENTAL RULE Q1. There are 4 bus lines between A & B and 3 bus lines between B & C. In how many ways can a man travel by bus from A to C by way of B. In how many ways can a man travel round trip (A to C and back). If he does not want to use a bus line more than once. Q2. How many words can be formed from letters of the word FATHER using all the alphabets? Q3. There are 9 different alphabets. How many words can you form using (a) 4 letters (b) 5 letters (c) 6 letters (d) 7 letters (e) 8 letters (f) 9 letters Q4. Find how many 4 digit numbers can be formed from the digits 0,1,2,3,4,5. Q5. How many numbers of 4 digit can be formed with digits 3, 5, 7, 8, 9. No digit being repeated. How many of these will be greater than 8000. PERMUTATIONS Permutation Means Arrangement Each of the different arrangement that can be made with a given number of objects by taking all or some of the objects at a time without repetition of any object is called permutation. The number of permutations of n different objects taken r (r ( n) at time will be denoted by the symbol nPr or P(n,r). There are two types of permutations (i) Linear Permutation (ii) Circular Permutations. (i) Linear Permutation: Permutations of n different objects taken r at a time is, the total number of ways in which n objects can be arranged at r places in a line and denoted by  EMBED Equation.3  or P(n,r) or nPr (a) The number of permutations of n different objects taken r at a time, when repetition of r objects in the permutation is not allowed is given by  EMBED Equation.3  =  EMBED Equation.3 Where r ( n. (b) The number of permutations of n different objects taken all at a time, when 2 specified objects always come together is 2(n-1)!. (c) The number of permutation of n different objects taken all at a time, when 2 specified objects never come together is (n-2) (n-1)!. (d) The number of permutations of n different objects taken r at a time, when particular object is to be always included in each arrangement is  EMBED Equation.3  (e) The number of permutations of n different objects taken r at a time, when particular object is never included in each arrangement is  EMBED Equation.3  (f) The number of permutations of n different objects taken r at a time, when repetition of objects in the permutation is allowed, is given by nr. Some important Results. 1. The number of permutations of n distinct objects taken r at a time, when repetition of objects is allowed, is nr (r>0). 2. The number of permutations of n distinct objects taken r (0 ( r ( n) at a time, when repetition is not allowed, is given by  EMBED Equation.3  Obviously, (i)  EMBED Equation.3  = 0 for r > n. (ii)  EMBED Equation.3  (iii)  EMBED Equation.3  SIMPLE PROBLEMS ON PERMUTATION WITH SOME RESTRICTION Q6. Indicate how many 4 digit numbers greater than 7000 can be formed from the digits 3, 5, 7, 8, 9. Q7. How many different numbers of six digits can be formed with the digits 3, 1, 7, 0 9, 5? How many of these have 0 in tens place. Q8. (i) How many odd numbers of 5 digit can be formed with the digit 3, 2, 7, 4, 0 when no digit is repeated ? (ii) How many even numbers can be formed? Q9. How many numbers between 3000 and 4000 can be formed with the digits 1, 2, 3, 4, 5. Q10. How many numbers each lying between 100 and 10,000 can be formed with the digits 1,2,3,5,7,9. No digit being repeated? Q11. A number of 4 different digits is to be formed by using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways. Find (i) How many such numbers can be formed ? (ii) How many of them will be greater than 3,400 ? Q12. How many numbers between 300 and 3000 can be formed with the digits 0,1,2,3,4, and 5 , no digit being repeated in any number Q13. Find numbers less than 1000 and divisible by 5 which can be formed with digits 0,1,2,3,4,5,6,7,8,9. No digit should occur more than once. Q14. There are 50 stations on the railway line. How many different kinds of single ticket must be printed so as to enable a passenger to go from one station to other? Q15. There are 8 vacant chairs in a room. In how many ways can 5 persons take their seats Q16. How many different words can be formed with the letters of the word SUNDAY. How many will begin with N and how many will end with Y? Q17. How many different arrangements can be made by using all the letters of the word MONDAY. How many of these arrangements begin with A and end with N. Q18. In how many ways 6 papers of an examination be arranged so that the worst and the best papers are always together? Q19. How many arrangements of the letters of the word COMRADE can be made such that (i) Vowels are never separated. (ii) Vowels are to occupy odd places. Q20. In how many ways can the letters of the word FAILURE be arranged so that vowels occupy only odd positions? Q21. Find how many words can be formed of the letters of the word FAILURE; the four vowels always come together. Q22. In how many ways can 3 boys and 5 girls be arranged in a row so that all the 3 boys are together. Q23. There are 6 books on Economics, 3 books on mathematics and 2 books on accountancy. In how many ways can these be placed on a shelf if the books on same subject are always together? Q24. In how many ways can 3 books on Mathematics, and 5 books on Secretarial practice be placed on a shelf so that books on the same object always remain together. Q25. How many different words containing all the letters of the word TRIANGLE can be formed? How many of them i) Begin with T ii) Begin with E iii) Begin with T and end with E iv) Have T and E at its end places v) When the consonants are never separated vi) When vowels always occupy odd places vii) Vowels occupy 2nd 3rd and 4th position. PERMUTATION OF THINGS NOT ALL DIFFERENT If n object are to be arranged amongst themselves, of which p are identical of one type, q are identical of another type, r identical of a third type and the rest are all different, then the number of permutations, when all n objects are arranged amongst themselves, is given by  EMBED Equation.3 . Q26. In how many ways can the letters of the word CONSTITUTION be arranged? Q27. In how many ways can the letters of the word MATHEMATICS be arranged? Q28. Find the number of ways that can be formed by the digits 1,2,3,4,3,2,1 by placing the odd digits at odd places. (Nov. 01) Q29. How many different words can be made out of the letters in the word ALLAHABAD. In how many of these the vowels will occupy even places? Q30. How many different words can be made out of the letters of the word CALCUTTA. In how many of these will the vowels occupy even places. Q31. If the letters of the word WOMAN be permuted and the words so formed be arranged in a dictionary. What will be the rank of woman? Q32. Find the number of ways in which the letters of the word ZENITH can be permuted. What will be the rank of word ZENITH if all the words appear in a dictionary? Example: Compute the sum of 4 digit numbers which can be formed with the four digits 1, 3, 5, 7, if each digit is used only once in each arrangement. Solution: The number of arrangements of 4 different digits taken 4 at a time is given by 4! = 24. All the four digits will occur equal number of times at each of the position, namely ones, tens, hundreds, thousands. Thus, each digit will occur 24/4 = 6 times in each of the position. The sum of digits in ones position will be 6 (1 + 3 + 5 + 7) = 96. Similar is the case in tens hundreds and thousands place. Therefore, the sum will be 96 + 96x10 + 96x100 + 96x1000 = 106656 Circular permutations: Circular arrangements or permutations are related to arrangement of objects as in case of sitting arrangement in a round table conference etc. In case of circular arrangement permutation is not different unless the order or relative positions of the objects change. A E B E A D C D B C Both of the above arrangements are same. If a circular arrangement of n objects is to be made we have to first fix one object at a particular place and the remaining objects can be fixed in (n - 1)! ways. Note: In case of sums where the objects are repeated on necklace , garland, it will be  EMBED Equation.3  Q33. In how many ways can 5 persons be seated at a round table conference? Q34. In how many ways can 7 gentleman and 7 ladies sit at a round table so that no two ladies are together? Q35. In how many ways can 4 men and 3 ladies be arranged at a round table if the 3 ladies (i) Never sit together (ii)Always sit together Q36. The chief minister of 18 states in India meet and discuss the problem of unemployment. In how many ways can they seat themselves at a round table if Punjab and West Bengal chief ministers choose to sit together? Q37. A round table conference is to beheld for a committee of 7 persons which includes president and secretary. Find the number of way the committee can be seated so that The president and secretary sit together The secretary sits to the right hand side of the president The president and secretary do not sit together COMBINATIONS Combination Mean Selection (or Forming Groups) A group or a selection which can be formed by taking some or all of a number of objects irrespective of their arrangements is called a combination. Here we are concerned only with the number of things in each selection and not with the order of things. The number of combinations (or selections) of a n different things taken r at a time (r ( n) is denoted by the symbol  EMBED Equation.3  or C(n,r). Notes: In permutations each arrangement is regarded as a unique arrangement Eg. While calculating permutations of 3 objects A, B and c even if the internal arrangement changes such as A, B, C C, B, A B, A, C etc. Each of the above arrangement is regarded as different. But sometimes we come across such a situation where order n of the items selected is not important. Such a situation is called problem of combination. For eg. If there are 52 cards and we have to select any 3. Such a problem is called problem of combination. Some important Results 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  (1 ( r ( n). 7. nC0 + nC1 + nC2 + + nCn = 2n. 8. nC0 + nC2 + .. = nC1+ nC3 + .. = 2n-1. 9. 2n+1C0 + 2n+1C1 + .. + 2n+ 1Cn = 22n 10. The number of combinations of n distinct objects taken r (( n) at a time, when k(0 ( k ( r) particulars objects always occur is n-k C r-k. 11. The number of combinations of n distinct objects taken r at a time, when k(1 ( k ( n-r)never occur, is n-kCr. Q38. In how many ways can a committee of 6 men and 2 women be formed out of 10 men and 5 women? Q39. In a college there are together 20 professors including the principal and vice principal from whom a committee of 5 is to be formed. Find out how many committees will contain The principal and vice principal. The principal but not the vice principal. (iii) Neither the principal nor the vice principal. Q40. Out of 4 officers and 10 clerks in an organisation, a committee of five consisting of 2 officers and 3 clerks is to be formed. In how many ways can this be done, if (i) any officer and any clerk can be included. (ii) any particular clerk must be on the committee. (iii) one particular officer cannot be on committee. Q41. Find out the number of ways in which a cricket team consisting of 11 players can be selected out of 14 players. Also find out how many of these will include a particular player. Q42. A student has to answer 8 out of 10 questions in an examination. (i) How many choices does he have. (ii) How many choices he has if he must answer 1st 3 questions. (iii) How many if he must answer atleast 4 questions out of the 1st 5 questions. Q43. A bag contains mixture of eight one rupee, six 50 paise and four 20 paise coins. In how many ways selections of 3 coins can be made such that (i) all 3 are rupee coins. (ii) one coin of each denomination is selected. (iii) none is a rupee coin. Q44. A box contains 7 red, 6 white and 4 blue balls, How many selections of three balls can be made so that a) None is red b) one is of each colour Q45. A Supreme Court Bench consists of five judges. In how many ways the bench can give a majority decision? Also calculate the ways of negative not affecting the majority decision. Q46. In an election a voter may vote for any number of candidates not greater than the number to be chosen. There are seven candidates and four members are to be chosen. In how many ways can a person vote? Q47. A council consists of 10 members. 6 belonging to party A and 4 belonging to party B. In how many ways can a committee of 5 be selected so that members of party A are in minority? Q48. A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket keepers. In how many different ways can a team be formed so that the team consists of at least 3 bowlers and at least one wicket keeper? Q49. There are 8 Gentlemen and 4 ladies, find the number of ways in which a committee of 7 members can be formed from these, if each committee is to include atleast 3 ladies Q50. An examination paper consists of questions divided into two parts A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to answer 8 questions selecting at least 3 questions from each part. In how many maximum ways can the candidate select the questions? Q51. A committee of 7 members is to be chosen from 6 C.A's, 4 Economists and 5 cost accountants. In how many ways can this be done if in the committee, there must be at least one member from each group and at least 3 C.A's? Q52. For a certain course of studies a student has to select 3 subjects out of 9 subjects. The subjects are divided into 3 groups each containing 3 subjects of which one is a practical subject. A student has to choose 1 subject from each group in such a way that atleast 1 but not more than 2 practical subjects are selected. In how many ways can this be done? Q53. A committee of 3 experts is to be selected out of panel of 7. 3 of them are lawyers and 3 are C.A.'s and 1 is both C.A and a lawyer. In how many ways can a committee be selected if there must be atleast 1 lawyer and 1 C.A.? Q54. Of 10 electric bulbs 3 are defective. But it is not known which are defective. In how many ways 3 bulbs can be selected. How many of these selections will include atleast one defective bulb? Q55. In the CA examinations candidate is required to pass in 4 different papers. In how many ways he can fail? Q56. In how many ways can a committee of 3 ladies and 4 gentleman be appointed from 8 ladies and 7 gentleman. What would be the number of ways if Mrs. X refuses to serve on a committee if Mr. Y is a member? Q57. In a multinational co. 3 branches in a particular country are to be managed by 4,5 and 8 persons respectively. In how many ways can 20 persons be allotted to different branches? Q58. There are 25 candidates which includes 5 from Scheduled Caste for 12 vacancies. If 3 vacancies are reserved for Scheduled caste candidates and the remaining vacancies are open to all, find the number of ways in which selection can be made. Q59. A firm of C.A's in Bombay has to send ten clerks to five concerns, two to each concern. Two of the concerns are in Bombay and the others are outside. Two of the clerks prefer to work in Bombay while three prefer to work outside. In how many ways can the assignment be made if preferences are to be satisfied? Q60. There are 15 points in a plane of which 5 are collinear. Find (i) The number of straight lines which can be obtained by joining these points in pairs. (ii) The number of triangles that can be formed with the vertices at this point. Note: Numbers of diagonals = nC2 n where n is number of vertices of a given polygon. Triangle = 4 sides, Quadrilateral = 4 sides, Pentagon = 5 sides, Hexagon = 6 sides, Heptagon = 7 sides, Octagon = 8 sides, Nonagon = 9 sides, Decagon = 10 sides PROBLEMS ON COMBINATION WITH REPITATION Q61. Find the number of combinations of the letters of the word COLLEGE taken four together Q62. Find the number of combinations that can be made by taking 4 letters of the word COMBINATION. Q63. Find the number of words of three letters that can be formed with the letters of the word CALCUTTA (a) 90 (b) 96 (c) 98 (d) None of these MULTIPLE CHOICE QUESTIONS (Theory) 1. Study about the problem of arranging and grouping of certain things, taking particular number of things at a time. (a) Permutations (b) Combinations (c) (a) or (b) (d) (a) and (b) 2. The ways of arranging or selecting smaller or equal number of persons or objects from a group o persons or collection of objects with due regard being paid to the order of arrangement or selection are known as (a) Permutations (b) Combinations (c) (a) or (b) (d) (a) and (b) 3. The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important, are known as (a) Permutations (b) Combinations (c) (a) or (b) (d) (a) and (b) 4. Number of permutations when r objects are chosen out of n different objects then it is denoted by (a) nPr (b) nPr (c) P(n, r) (d) Any of the above. 5. Under Linear permutations, arrangements of object or things in a ________. (a) row (b) column (c) curve (d) none 6. Under _______, arrangements of objects or things along a close curve. (a) Linear Permutations (b) Curve Permutations (c) Circular Permutations (d) Parabolic Permutations. 7. Number of _______ of n distinct objects when a particular object is not taken in any arrangement is r. n-1Pr. (a) Permutations (b) Combination (c) Circular Permutation (d) None. 8. In nPr, n is always n ( r. (a) True (b) Partly True (c) False (d) none 9. In nPr = n(n-1) (n-2) .. (n r -1), the number of factor is n. (a) True (b) Partly True (c) False (d) none 10. nPr and  EMBED Equation.3  is the same, (a) True (b) Partly True (c) False (d) none 11. n articles are arranged in such a way that two particular articles never come together. The number of such arrangements is (n-2)  EMBED Equation.3  (a) True (b) Partly True (c) False (d) none 12. Number of circular permutations of n different things chosen at a time is  EMBED Equation.3  (a) True (b) Partly True (c) False (d) none 13. The number of ways of arranging n persons along a round table so that no person has the same two neighbors is  EMBED Equation.3  (a) True (b) Partly True (c) False (d) none 14. Number of permutation of n distinct objects when a particular object is not taken in any arrangement is nPr-1. (a) True (b) Partly True (c) False (d) none 15. Number of permutations of n distinct objects when a particular object is always included in any arrangement is r. n-1Pr ways. (a) True (b) Partly True (c) False (d) none 16. (n-1)Pr + r(n-1)P(r-1) is equal to nPr. (a) True (b) Partly True (c) False (d) none 17. nC1 + nC2 + nC3 +..+ equals to 2n. (a) True (b) Partly True (c) False (d) none Answers: 1.(d) 2.(a) 3.(b) 4.(d) 5.(a) 6.(c) 7.(a) 8.(a) 9.(c) 10.(a) 11.(a) 12.(a) 13.(a) 14.(c) 15.(a) 16.(a) 17.(c) MULTIPLE CHOICE QUESTIONS (Practical) 1. The letters of the words CALCUTTA and AMERICA are arranged in all possible ways. The ratio of the number of these arrangements is _______ a. 1:2 b. 2:1 c. 1:1 d. 1.5:1 2. There are 10 trains playing between Calcutta and Delhi. The number of ways in which a person can go from Calcutta to Delhi and return by a different train is a. 99 b. 90 c. 80 d. None of these 3. Every person shakes hands with each other in a party and the total number of hand shakes is 66. The number of guests in the party is a. 11 b. 12 c. 13 d. 14 4. 4P4 is equal to a. 1 b. 24 c. 0 d. None of these 5. In _______ ways can 4 Americans and 4 English men be seated at a round table so that no 2 Americans may be together a. 4( x 3( b. 4P4 c. 3 x 4P4 d. 4C4 6. Find n if nP3 = 60 a. 4 b. 5 c. 6 d. 7 7. A man has 5 friends. In how many ways can he invite one or more of his friends to dinner ? a. 29 b. 30 c. 31 d. 32 8. If there are 50 stations on a railway line how many different kinds of single first class tickets may be printed to enable a passenger to travel from one station to other? a. 2500 b. 2450 c. 2400 d. None of these 9. If 5Pr = 60, then the value of r is a. 3 b. 2 c. 4 d. None of these 10. A question paper contains 6 questions, each having an alternative. The number of ways an examiner can answer one or more questions is a. 720 b. 728 c. 729 d. None of these 11. Eleven students are participating in a race. In how many ways the first 5 prizes can be won ? a. 44550 b. 55440 c. 120 d. 90 12. Find the value of n if (n+1)! = 42(n-1)! a. 6 b. -7 c. 7 d. -6 13. Compute the value of 8! a. 120 b. 362880 c. 720 d. 40320 14. Find the value of 8!/5! a. 663 b. 363 c. 336 d. None of these 15. A letter lock has three rings each marked with 10 different letters. In how many ways it is possible to make an unsuccessful attempt to open the lock ? a. 1000 b. 999 c. 5040 d. None of these 16. How many numbers greater than 2000 can be formed with the digits 1,2,3,4,5? a. 216 b. 120 c. 24 d. 240 17. In how many ways can 4 single seated rooms in a hostel be occupied by 3 students ? a. 24 b. 12 c. 4 d. 6 18. No. of _______ arrangement can be made by using all the letters of word Monday. a. 120 b. 720 c. 41 d. 51 19. The number of ways in which 6 boys sit in a round table so that two particular boys sit together a. 48 b. 720 c. 120 d. None of these 20. In how many ways can the letters of words ACCOUNTANT be arranged if vowels always occur together a. 7560 b. 7650 c. 7660 d. 7550 21. How many diagonals can be drawn in a plane figure of 16 sides. a. 100 b. 50 c. 104 d. 54 22. If c(n,8) = c(n,6), find c(n,2) a. 14 b. 91 c. 19 d. 41 23. The total no. of seating arrangement of 5 person in a row is _______ a. 5! B. 4! C. 2x5! D. None of these 24. There are 11 trains plying between Delhi & Kanpur. The number of ways in which a person can go from Delhi to Kanpur and return by a different train a. 121 b. 100 c. 110 d. None of these 25. If in a party every person gives a gifts to each other and total number of gift taken is 132. The number of guests in the party is a. 11 b. 12 c. 13 d. 14 26. In how many ways can 7 persons be seated at a round table if 2 particular persons sit together a. 420 b. 1440 c. 240 d. None of these 27. The letters of the words ALLAHABAD and INDIA are arranged in all possible ways. The ratio of the number of these arrangement is _______ a. (9:(5 b. 126:1 c. 1:1 d. 2:5 28. If nP13:n+1P12 = 3:4 then value of n is a. 15 b. 14 c. 13 d. 12 29. In how many ways 5 physics, 3 chemistry and 3 Maths books be arranged keeping the books of the same subject together. a. 5!x3!x3! b. 5P3 c. 5!x3! d. 5!x3!x3!x3! 30. 6 seats of articled clerks are vacant in a Chartered Accountant firm. How many different batches of candidates can be chosen out of 10 candidates if one candidate is always selected. a. 124 b. 125 c. 126 d. None of these 31. How many three digit numbers are there, with distinct digits, with each digits odd a. 120 b. 60 c. 30 d. 15 32. In how many ways can the letters of the word PENCIL be arranged so that N is always next to E a. 60 b. 40 c. 720 d. 120 33. The no. of permutation can be made out the letters of words COMMERCE IS ________ a. 5040 b. 8! C. 6! D. None of these 34. How many words can be formed out of 5 different consonants and 4 different vowels if each word is to contain 3 consonants and 2 vowels a. 7000 b. 720 c. 7020 d. 7200 35. 7 distinct things are to be divided in 3 groups, consisting of 2, 2 and 3 things respectively, no. of ways this can be done is equal to _________ a. 110 b. 210 c. 100 d. None of these 36. In how many ways 5 gents and 5 ladies sit at a round table; if no two ladies are to sit together a. 720 b. 120 c. 2,880 d. 34,600 37. Out of 6 teachers and 4 boys, a committee of eight is to be formed. In how many ways can this be done when there should not be less than four teachers in the committee. a. 45 b. 55 c. 30 d. 50 38. (0 x (6 is equal to a. 720 b. 0 c. 6 d. -120 39. In how many ways can 8 persons sit at a round table for a meeting ? a. 40320 b. 64 c. 5040 d. 720 40. There are 7 routes from station X to station Y. In how many ways one may go from X to Y and return if for returning one makes a choice of any of the routes ? a. 49 b. 17 c. 42 d. 35 41.  EMBED Equation.3  is equal to a. 60 b. 0 c. 120 d. None of these 42. (0 x (7 x (2 is equal to _______ a. 10080 b. 0 c. 5040 d. None of these 43. The value of 11p9 is equal to a. EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d. None of these 44. How many different numbers can be formed by using any four out of six digits 1,2,3,4,5,6, no digit being repeated in any number ? a. 60 b. 120 c. 30 d. 360 45. How many five digit numbers can be formed out of digits 1,2,4,5,6,7,8, if no digit is repeated in any number ? a. 2502 b. 2520 c. 2205 d. None of these 46. A committee of 7 persons is to be formed out of 11. The number of ways of forming such as committee is _______ a. 660 b. 330 c. 300 d. None of these 47. How many different arrangements are possible from the letters of the word CALCULATOR ? a. 453600 b. 50400 c. 45360 d. None of these 48. A man has 7 friends. In how many ways can he invite one or more of his friends ? a. 127 b. 256 c. 255 d. None of these 49. There are 7 boys and 3 girls. The number of ways, in which a committee of 6 can be formed from them, if the committee is to include at least 2 girls is ________ a. 140 b. 105 c. 35 d. None of these 50. 5C1 + 5C2 + 5C3 + 5C4 + 5C5 is equal to _______ a. 30 b. 31 c. 32 d. 25 51. The sum of all 4 digit number containing the digits 2,4,6,8, without repetitions is (a) 133330 (b) 122220 (c) 213330 (d) 133320 52. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is (a) 6 (b) 18 (c) 12 (d) 9 53. The number of diagonals that can be drawn by joining the angular points of a heptagon is: (a) 21 (b) 14 (c) 7 (d) 28. 2006 November 1. The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is: a) 185 b) 175 c) 115 d) 105 2. A code word is to consist of two English alphabets followed by two distinct number between 1 and 9. How many such code words are there? a) 6,15,800 b) 46,800 c) 7,19,500 d) 4,10,800 3. A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics part II unless Mathematics part I is also borrowed? In how many ways can he choose the three books to be borrowed? a) 41 b) 51 c) 61 d) 71 2007 February 1. An examination paper consists of 12 questions divided into two parts A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions selecting at least 3 from each part. In how many maximum ways can the candidate select the questions? a) 35 b) 175 c) 210 d) 420 2. A Supreme Court Bench consists of 5 judges. In how many ways, the bench can give a majority division? a) 10 b) 5 c) 15 d) 16 3. Given: P (7,k) = 60 P(7,k-3). Then: a) k = 9 b) k = 8 c) k = 5 d) k = 0 4. The number of ways in which n books can be arranged on a shelf so that two particular books are not together is: a) (n-2) x (n-1)! b) (n-2) x (n+1)! c) (n-1) x (n+1)! d) (n-2) x (n+2)! 2007 May 1. In how many ways can the letters of the word FAILURE be arranged so that the consonants may occupy only odd positions? a) 576 b) 476 c) 376 d) 276 2. Five bulbs of which three are defective are to be tried in two lights- points in a dark-room. In how many trials the room shall be lighted? a) 10 b) 7 c) 3 d) none 3. In how many ways can a party of 4 men and 4 women be seated at a circular table, so that no two woman are adjacent? a) 164 b) 174 c) 144 d) 154 4. The value of  EMBED Equation.3  a) 29 b) 31 c) 35 d) 26 2007 August 1. If 6Pr = 246Cr, then find r: a) 4 b) 5 c) 6 d) 7 2. Find the number of combinations of the letters of the word COLLEGE taken four together: a) 18 b) 16 c) 20 d) 26 3. How many words can be formed with the letters of the word ORIENTAL so that A and E always occupy odd places: a) 540 b) 8640 c) 8460 d) 8450 2007 November 1. If1000C98 = 999C97 + xC901, find x: a) 999 b) 998 c) 997 d) 1000 2. How many numbers greater than a million can be formed with the digits 4,5,5,0,4,5,3? a) 260 b) 360 c) 280 d) 380 3. A building contractor needs three helpers and ten men apply. In how many ways can these selections take place? a) 36 b) 15 c) 150 d) 120 2008 February 1. There are three blue balls, four red balls and five green balls. In how many ways can they be arranged in a row? a) 26720 b) 27720 c) 27820 d) 26,620 2. If C(n,r) :C(n,r+1) = 1:2 and C(n,r+ 1): C(n,r+2) = 2:3, determine the value of n and r: a) (14,4) b) (12,4) c) (14,6) d) None. 2008 June 1. Six seats of articled clerks are vacant in a Chartered Accountant Firm. How many different batches of candidates can be chosen out of ten candidates? a) 216 b) 210 c) 220 d) none 2. Six persons A,B,C,D,E and F are to be seated at a circular table. In how many ways can this be done, if A must always have either B or C on his right and B must always have either C or D on his right? a) 3 b) 6 c) 12 d) 18 2008 December 1. If  EMBED Equation.3  and  EMBED Equation.3  then find the value of `n a) 2 b) 3 c) 4 d) 5 2. How many six digit telephone numbers can be formed by using 10 distinct digits? a) 106 b) 610 c)  EMBED Equation.3  d)  EMBED Equation.3  3. In how many ways a committee of 6 members can be formed from a group of 7 boys and 4 girls having at least 2 girls in the committee. a) 731 b) 137 c) 371 d) 351 2009 June 1. Number of ways of painting a face of a cube by 6 colours is_______ a) 36 b) 6 c) 24 d) 1 2. If ________ 18 Cr = 18 Cr + 2 Find the value of rC5. a) 55 b) 50 c) 56 d) none of these 3. 7 books are to be arranged in such a way so that two particular books are always at first and last place. Final the number of arrangements. a) 60 b) 120 c) 240 d) 480 4. Find the number of arrangements in which the letters of the word MONDAY be arranged so that the words thus formed begin with M and do not end with N. a) 720 b) 120 c) 96 d) none 5. In how many ways can 17 billiard balls be arranged if 7 of them are black, 6 red and 4 white? a) 4084080 b) 1 c) 8048040 d) none of these 2009 December 1. (n+ 1)! = 20 (n-1)!, find n (a) 6 (b) 5 (c) 4 (d) 10 2. Out of 4 gents and 6 ladies, a committee is to be formed find the number of ways the committee can be formed such that it comprises of at least 2 gents and at least the number of ladies should be double of gents. (a) 94 (b) 132 (c) 136 (d) 104 3. In how many ways can the letters of REGULATION be arranged so that the vowels come at odd places? (a) 1/252 (b) 1/144 (c) 144/252 (d) None of these 2010 June 1. Six points are on a circle. The number of quadrilaterals that can be formed are: (a) 30 (b) 360 (c) 15 (d) None of the above 2. The number of ways of arranging 6 boys and 4 girls in a row so that all 4 girls are together is: (a) 6!. 4! (b) 2 (7!.4!) (c) 7!. 4! (d) 2. (6!.4!) 3. How many numbers not exceeding 1000 can be made from the digits 1,2,3,4,5,6,7,8,9 if repetition is not allowed. (a) 364 (b) 585 (c) 728 (d) 819 2010 December 1. A garden having 6 tall trees in a row. In how many ways 5 children stand, one in a gap between the trees in order to pose for a photograph? (a) 24 (b) 120 (c) 720 (d) 30 2. 15C3 + 15C13 is equal to: (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d)  EMBED Equation.3  3. How many ways a team of 11 players can be made out of 15 players if one particular player is not to be selected in the team. (a) 364 (b) 728 (c) 1,001 (d) 1,234 2011 June 1. Find the number of arrangements of 5 things taken out of 12 things, in which one particular thing must always be included. (a) 39,000 (b) 37,600 (c) 39,600 (d) 36,000 2. Exactly 3 girls are to be selected from 5 Girls and 3 Boys. The probability of selecting 3 Girls will be _______. (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d) None 2011 December 1. In how many ways 3 prizes out of 5 can be distributed amongst 3 brothers Equally? (a) 10 (b) 45 (c) 60 (d) 120 2. There are 12 question to be Answered to be Yes or No. How many ways can these be Answered? (a) 1024 (b) 2048 (c) 4096 (d) None 3. A team of 5 is to be selected from 8 boys and three girls. Find the probability that it includes two particular girls. (a) 2/30 (b) 1/5 (c) 2/11 (d) 8/9 2012 June 1. In a company there are 7 CAs; 6 M.B.As and 3 engineers. How many ways can they form a committee, If there two members from each filed is (a) 900 (b) 1000 (c) 787 (d) 945 2. The letters of the word VIOLENT are arranged so that vowels occupy even places only. The number of permutations is (a) 144 (b) 120 (c) 24 (d) 72 2012 - December 1. A man has 3 sons and 6 schools within his reach. How many ways can his Sons go to school, if no two of them are in same school. (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d) 3 EMBED Equation.3  2. if  EMBED Equation.3   EMBED Equation.3 ,then x=---- (a) 6 (b) 7 (c) 8 (d) 9 3. Number of permutations can be formed from the letters of the word DRAUGHT If no two vowels are separable. (a) 720 (b) 1440 (c) 140 (d) 2880 2013 - June 1. The total number of shake hands in a group of 10 persons to each other are____ a) 45 b) 54 c) 90 d) 10 2. A regular polygon has 44 diagonals then the No. of sides are___ a) 8 b) 9 c) 10 d) 11 3. in how many ways the word ARTICLE can be Arranged in a row so that vowels occupy even Places? a) 132 b) 144 c) 72 d) 160 2013- December 1. How many different words can be formed with letters of the word LIBERTY (a) 4050 (b) 5040 (c) 5400 (d) 4500 2. In how many ways can a family consist of 3 children have different birthdays in a leap year (a) 366 x 365 x 364 (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d)  EMBED Equation.3  3. If  EMBED Equation.3  then r = (a) 2 (b) 3 (c) 4 (d) 5 2014- June 1. If 6 times the no. of permutations of n items taken 3 at a time is equal to 7 times the no. of permutations of (n-1) items taken 3 at a time then the value of n will be (a) 7 (b) 9 (c) 13 (d) 21 2. If1000C98 = 999C97 + xC901, find x: (a) 999 (b) 998 (c) 997 (d) none 2014- Dec 1.  EMBED Equation.3  =360 then find r (a) 4 (b) 5 (c) 6 (d) none 2. If 5 books of English, 4 books of Tamil and 3 books of Hindi are to be arranged in a single row so that books of same language come together (a) 1,80,630 (b) 1,60,830 (c) 1,03,680 (d) 1,30,680 3. 5 boys and 4 girls are to be seated in row. If the girls occupy even places then the no. of such arrangements (a) 288 (b) 2808 (c) 2008 (d) 2880 2015. June 1. A person has 10 friends of which 6 of them are relatives. He wishes to invite 5 persons so that 3 of them are relatives. In how many ways he can invites? (a) 450 (b) 600 (c) 120 (d) 810 2. A student has 3 books on computer, 3 books on Economics, 5 on Commerce. If these books are to be arranged subject wise then these can be placed on a shelf in the ________ number of ways. (a) 25, 290 (b) 25,920 (c) 4,230 (d) 4,320 3. The number of 4 digit numbers that can be formed from seven digits 1,2,3,5,7,8,9 such that no digit being repeated in any number, which are greater than 3000 are (a) 120 (b) 480 (c) 600 (d) 840 2015 Dec 1. A questions paper consist 10 questions, 6 in math and 4 in stats. Find out number of ways to solve question paper if at least one question is to be attempted from each section. (a) 1024 (b) 1023 (c) 945 (d) None of these 2. There are 6 gents and 4 ladies. A committee of 5 is to be formed if it include at least two ladies. (a) 64 (b) 162 (c) 102 (d) 186 3. nPr = 720 and nCr = 120 find r? (a) 6 (b) 4 (c) 3 (d) 2 2016- June 1. There are 10 students in a class, including 3 girls. The number of ways arrange them in a row, when any two girls out of them never come together (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d) none 2. In how many ways can a selection of 6 out of 4 teachers and 8 students be done so as to include atleast two teachers? (a) 220 (b) 672 (c) 896 (d) 968 3. The maximum number of points of intersection of 10 circles will be (a) 2 (b) 20 (c) 90 (d) 180 2016 Dec. 1. How many numbers between 1000 and 10,000 can be formed with the digits 1,2,3,4,5,6 (a) 720 (b) 360 (c) 120 (d) 60 2. If  EMBED Equation.3  Then find the value of n (a) 14 (b) 15 (c) 16 (d) 17 3. In how many ways 4 members can occupy 9 vacant seats in a row (a) 3204 (b) 3024 (c) 49 (d) 94 2017 June 1. The number of arrangements that can be formed from the letters of the word ALLAHABAD (a) 7560 (b) 3780 (c) 30240 (d) 15320 2. If  EMBED Equation.3  +  EMBED Equation.3  +  EMBED Equation.3  =  EMBED Equation.3  then the value of n = ________ (a) 10 (b) 11 (c) 12 (d) 13 3. The number of parallelograms that can be formed by a set of 6 parallel lines intersected by the another set of 4 parallel lines is ______ (a) 360 (b) 90 (c) 180 (d) 45 2017 Dec. 1. If nP13:(n+1) P12 = 3:4 then n is _______ (a) 13 (b) 15 (c) 18 (d) 31 2. In how many ways that 3 commerce books, 3 computer books and 5 economics books be arranged along a row, so that books of same subjects are come together is ______ (a) 25,950 (b) 25,940 (c) 25,920 (d) None 2018 JUNE 1. If 12C3 +2 (12C4) + 12C5 = 14Cx. What is x? (a) 3 or 5 (b) 5 or 9 (c) 7 or 1 (d) 9 or 12 2. The number of ways in which a man can invite one or more of his 7 friends to dinner is? (a) 64 (b) 128 (c) 127 (d) 63 3. If the pth, qth & rth term of a G.P. are x, y, z then (q r). log x + (r p) log y + (p q) log z = (a) 0 (b) 1 (c) 2 (d) None of these 4. If a, b, c, d are in GP then (b c)2 + (c a)2 + (d b)2 = ? = (a) (a b)2 (b) (a- d)2 (c) (c-d)2 (d) 0 5. If the nth term of a series, tn = 3n - 2n then Sn = (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  (d)  EMBED Equation.3  Questions for Practice 1. The letter of the words CALCUTTA and AMERICA are arranged in all possible ways. The ratio of the number of these arrangements is ___________. a) 1 : 2 b) 2 : 1 c) 1 : 1 d) 1.5 : 1 2. There are 10 trains plying between Calcutta and Delhi. The number of ways in which a person can go from Calcutta to Delhi and return by a different train is a) 99 b) 90 c) 80 d) None of these 3. Every person shakes hands with each other in a party and the total hand hakes is 66. The number of guests in the party is a) 11 b) 12 c) 13 d) 14 4.  QUOTE   is equal to a) 1 b) 24 c) 0 d) None of these 5. In how many ways can 8 persons be seated at a round table? a) 5040 b) 4050 c) 450 d) 540 6. Five bulbs of which three are defective are to be tired in two bulb points in a dark room. Number of trails the room shall be lighted a) 6 b) 8 c) 5 d) 7 7. In ________ ways can 4 Americans and 4 English men be seated at a round table so that no 2 Americans may be together. a) 4!  QUOTE   3! b)  QUOTE   c) 3  QUOTE   d)  QUOTE   8. Find n if  QUOTE   a) 4 b) 5 c) 6 d) 7 9. A man has 5 friends. In how many ways can he invite one or more of his friends to dinner? a) 29 b) 30 c) 31 d) 32 10. If  QUOTE   = 60, then the value of r is a) 3 b) 2 c) 4 d) None of these 11. A question paper contains 6 questions, each having an alternative. The number of ways an examiner can answer one or more questions is a) 720 b) 728 c) 729 d) None of these 12. A person has 8 friends. The number of ways in which he may invite one or more of them to a dinner are a) 250 b) 255 c) 200 d) None of these 13. Eleven students are participating in a race. In how ways the first 5 prizes can be won? a) 44550 b) 55440 c) 120 d) 90 14. Find the value of n if (n + 1)! = 42(n 1)! a) 6 b) 7 c) 7 d) 6 15. If  QUOTE   then value of n is _________. a) 5 b) 2 c) 1 d) 3 16. Find the value of  QUOTE   a) 663 b) 363 c) 336 d) None of these 17. A letter lock has three rings each marked with 10 different letters. In how many ways it is possible to make an unsuccessful attempt to open the lock? a) 1,000 b) 999 c) 5,040 d) None of these 18. Value of  QUOTE   is _______ a) 6 b) 1 c) 3 d) 2 19. In how many ways can 4 single seated rooms in a hostel be occupied by 3 students? a) 24 b) 12 c) 4 d) 6 20. Compute  QUOTE   a) 8 b) 7 c) 6 d) None of these  QUOTE   can be expressed as a)  QUOTE   b)  QUOTE   c)  QUOTE   d) None of these 22. How many numbers greater than 2000 can be formed with the digits  QUOTE   with each digit distinct? a) 216 b) 120 c) 24 d) 240 23. Compute the value of 8! a) 120 b) 362880 c) 720 d) 40320 24. If there are 50 stations on a railway line how many different kinds of single first class tickets may be printed to enable a passenger to travel from one station to another? a) 2500 b) 2450 c) 2400 d) None of these 25. The ways of selecting 4 letters from the EXAMINATION is a) 136 b) 130 c) 125 d) None of these 26. In how many ways 3 letters can be formed using the letters of the words SPECIAL? a) 210 b) 6 c) 840 d) 450 27. The value of  QUOTE   is a) 14 b) 24 c) 30 d) 27 28. If  QUOTE   then the value of n is ___________. a) 0 b) 2 c) 8 d) None of above 29. How many numbers between 100 and 1000 can be formed with the digits. 2,3,4,0,8,9? (a) 100 (b) 105 (c) 200 (d) None of these 30. In how many ways can the letter of the word ALGEBRA be arranged without changing the relative order of the vowels? a) 82 (b) 70 (c) 72 (d) None of these 31. How many words can be formed with the letters of the word UNIVERSITY, the vowels always remaining together? a) 60480 (b) 60482 (c) 60000 (d) None of these 32. In how many ways can the letters of the word DIRECTOR be arranged so that the three vowels are never together? (a) 180 (b) 18,000 (c) 18,002 (d) None of these 33. A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and 1 Chartered Accountant. Find the chance there must be one from each category. (a) 4/35 (b) 3/35 (c) 1/7 (d) None of these 34. A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and 1 Chartered Accountant. Find the chance that it should have at least one from the purchase department. (a) 4/35 (b) 39/42 (c) 42/105 (d) None of these 35. A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and 1 Charted Accountant. Find the chance that the Chartered Accountant must be in the committee. (a) 4/35 (b) 39/42 (c) 42/105 (d) None of these 36. Six boys and five girls are to be seated in a row such that no two girls and no two boys sit together. Find the number of ways in which this can be done. (a) 86,400 (b) 85,000 (c) 85,400 (d) none of these 37. The number of ways in which n different books can be arranged in an almirah so that two particular books are always together is - a) n! x 2! (b) (n 1)! X 2! (c) (n 2)! (d) None 38. There are 3 copies each of two books and two copies each of five books. In how many ways can a book seller arrange the 16 books in a shelf so that the copies of the same book are never separated? (a) 5040 (b) 5000 (c) 5030 (d) None 39. How many words can be formed with the letters of the world PARALLEL so that all Ls do not come together? (a) 2000 (b) 3000 (c) 4000 (d) None 40. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is (a) 6 (b) 18 (c) 12 (d) 9 41. Three groups of children contain respectively 3 girls and 1 boys; 2 girls and 2 boys; 1 girl and 3 boys. One child is selected at random from each group. Then the chance that the three selected consist of 1 girl and 2 boys is: (a) 17/32 (b) 15/32 (c) 13/32 (d) None 42. If 56 Pr+6 : 54 Pr+3 = 30800 :1 then the value of r is (a) 42 (b) 41 (c) 45 (d) None of these 43. When John arrives in New York, he has eight shops to see, but he has times only to visit six of them. In how many different ways can he arrange his schedule in New York? (a) 20000 (b) 20160 (c) 21160 (d) None 44. There are 6 students of whom 2 are Indians, 2 Americans, and the remaining 2 are Russians. They have to stand in a row for a photograph so that the two Indians are together, the two Americans are together and so also the two Russians., Find the number of ways in which they can do so. (a) 40 (b) 42 (c) 48 (d) None of these 45. Find the number of different poker hands in a pack of 52 playing cards. (a) 2598960 (b) 1506210 (c) 5298216 (d) None 46. In how many different ways can I invite one or more of my 6 friends? (a) 63 (b) 64 (c) 60 (d) None of these 47. In an examination a candidate has to pass in each of the 4 papers. In how many different ways can be failed? (a) 14 (b) 16 (c) 15 (d) None of these 48. In an election the number of candidates is one more than the number of members to be elected. If a voter can vote in 254 different ways; find the number of candidates. (a) 8 (b) 10 (c) 7 (d) None of these 49. From 17 consonants and 5 vowels, how many words of 3 consonants and 2 vowels can be made if all the letters are different? (a) 810000 (b) 816000 (c) 815000 (d) None 50. A boat is to be manned by 8 men of which 3 can row only one side and 2 only on the other. In how many ways can the crew be arranged if equal number of men sit on the both sides. (a) 1720 (b) 1700 (c) 1728 (d) None of these 51. Three gentlemen and three ladies are candidates for two vacancies. A voter has to vote for two candidates. In how many different ways can one cast his vote? (a) 10 (b) 12 (c) 15 (d) None of these 52. In a party of 40 people, each shakes hand with others. How many handshakes took place in the party? (a) 780 (b) 700 (c) 880 (d) None 53. How many different triangles can be formed joining the angular points of a polygon of m sides? (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c) m (d) None 54. How many different cricket teams of 11 players can be selected from 14 cricket players of which only two can play as wicketkeeper? Given each team must have exactly one wicketkeeper? (a) 130 (b) 132 (c) 140 (d) None of these 55. Mr. X has 8 children of which he takes 3 at a time to the circus. Find, how many times a particular child goes to the circus? (a) 20 (b) 30 (c) 21 (d) None of these 56. If 12C5 + 2 12C4 + 12C3 = 14Cx then the value of x is: (a) 5 (b) 9 (c) 5 or 9 (d) None of these 57. There are 7 man and 3 ladies. Find the numbers of ways in which a committee of 6 can be formed of them if the committee is to include atleast two ladies. (a) 140 (b) 130 (c) 105 (d) None 58. If 28C2r : 24C2r-4 = 225 : 11, then the value of r is (a) 10 (b) 7 (c) 5 (d) None of these 59. A committee is to be formed of 3 persons out of 12. Find the number of ways of forming such a committee. (a) 210 (b) 230 (c) 220 (d) None 60. A gentleman invites 6 of his friends to a party. In how many different arrangements they along with the wife of the gentlement can sit at a round table for a dinner if the host and his wife always sit side by side? (a) 1440 (b) 144 (c) 1445 (d) None 61. In how many ways can 7 departments be distributed among 3 ministers, if every ministers gets at least one but not more than 3 departments? (a) 1050 (b) 1000 (c) 1200 (d) None 62. Find the number of words of three letters that can be formed with the letters of the word CALCUTTA (a) 90 (b) 96 (c) 98 (d) None of these 63. How many number of numbers three digits can be made from the digits of the number 1,2,3,4,3,2? (a) 40 (b) 42 (c) 45 (d) None of these 64. If nP3 = 60, then the value of n is (a) 3 (b) 10 (c) 5 (d) None of these 65. From a panel of 4 doctors, 4 officers and one doctor who is also an officer, how many committee of 3 can be made if it has to contain at least one doctor and one officer? (a) 76 (b) 78 (c) 80 (d) None of these 66. In an election, there are five candidates contesting for three vacancies; an elector can vote any number of candidates not exceeding the number of vacancies. In how many ways can one cast his votes? (a) 12 (b) 14 (c) 25 (d) None of these 67. Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all the five balls. In how many different ways can we place the balls so that no box remains empty? (a) 100 (b) 120 (c) 150 (d) None of these 68. The value of 33 +43 + 53 +.. + 113 (a) 4356 (b) 4348 (c) 4347 (d) 4374 69. If nP5 : nP3 = 2 : 1; then the value of n is (a) 4 (b) 5 (c) 10 (d) None of these 70. A room has 10 doors. In how many ways can a man enter the room by one door and come out by a different door. (a) 90 (b) 100 (c) 50 (d) None of these 71. How many numbers greater than 1000 can be formed with the digits of the number 23416; if the digits are not repeated in the same number. (a) 120 (b) 200 (c) 240 (d) None of these 72. How many numbers can be formed with the digits of the number 112321 that are greater than one lakh? (a) 60 (b) 80 (c) 70 (d) None of these 73. In how many different ways can 17 billiard balls be arranged, if 7 of them are black, 6 red and 4 white. (a) 408408 (b) 4084080 (c) 4004080 (d) None Answers. 1.(b) 2.(b) 3.(b) 4.(b) 5.(a) 6.(d) 7.(a) 8.(b) 9.(c) 10.(a) 11.(b) 12.(b) 13.(b) 14.(a) 15.(a) 16.(c) 17.(b) 18.(a) 19.(a) 20.(a) 21.(b) 22.(a) 23.(d) 24.(b) 25.(a) 26.(a) 27.(a) 28.(c) 29.(a) 30.(c) 31.(a) 32.(d) 33.(a) 34.(b) 35.(c) 36.(a) 37.(b) 38.(a) 39.(b) 40.(b) 41.(c) 42.(b) 43.(b) 44.(c) 45.(a) 46.(a) 47.(c) 48.(a) 49.(b) 50.(c) 51.(c) 52.(a) 53.(a) 54.(b) 55.(c) 56.(c) 57.(a) 58.(b) 59.(c) 60.(a) 61.(a) 62.(d) 63.(d) 64.(c) 65.(a) 66.(c) 67.(c) 68. 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