Permutations and Combinations Exercise

PERMUTATION AND COMBINATION Q.1 Page # 6 EXERCISE Compute: 9! (i) (5!)  (3!) (ii) 32! 29! (iii) students each, in XI standard. In how many ways can a set of 4 student representatives be chosen, one from each section ? (12!)  (10!) 9! Q.2 Find LCM [3!, 6!, 8!]. Q.14 Q.3 Write the following products in factorial notation: (i) 5  6  7  8  9  10  11  12 (ii) 3  6  9  12  15 In how many ways can a vowel, a consonant and a digit chosen out of the 26 letters of the English alphabet and the 10 digits ? Q.15 In a class there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class in a function. In how many ways can the teacher make this selection ? Q.16 Given A = {2, 3, 5} and B = {0, 1}. Find the number of different ordered pairs in which the first entry is an element of A and the second is an element of B. Q.17 How many arithmetic progressions with 10 terms are there whose first terms is in the set {1 , 2, 3} and whose common difference is in the set {2, 3, 4}? Q.18 There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible (correct or incorrect) answers are there to this question ? Q.19 There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choice each and the next three have 5 each? Q.20 A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all possibilities in the future years? Q.21 In a city, telephone numbers consist of 6 digits and none of them begins with 0. How many telephone numbers could be possible in that city? Q.22 In how many ways can 6 letters be posted in 5 letters boxes available in the locality? Q.23 How many 3–digits number of distinct digits can be formed from the digits 2, 3, 7, 8, 9? Q.24 Find the number of all even 2-digit numbers, not having 0 at the unit’s place. Q.25 How many 3-digit numbers are there, with distinct digits, with each digit odd ? Q.4 Prove that : (n !)  (n + 2) = [(n !)+ (n + 1) !]. Q.5 Which of the following are true ? (i) (2 + 3) ! = 2! + 3! (ii) (2 × 3) ! = (2!) × (3!) Q.6 Evaluate: n! , (r!)  (n  r)! when n = 15 and r = 12. Q.7 Find n, if : (i) (n + 2) ! = 60 × (n – 1) ! (ii) (n + 2) ! = 2550 (n !) Q.8 Prove that: n! = n(n  1)(n  2)….(r + 1) r! n! n! (ii) (n  r + 1)  = (n  r  1)! (n  r )! (i) Q.9 n! If 2  n  2 ! and n! 4! n  4 ! are in the ration 2 : 1. find n. Q.10 There are 10 buses running between Delhi and Agra. In how many can a man go from Delhi to Agra and return by a different bus ? Q.11 There are three routes: air, rail and road for going from Madras to Hyderabad. But from Hyderabad to Vikarabad, there are two routes, rail and road. How many kinds of routes are there from Madras to Vikarabad via Hyderabad Q.12 Q.13 In a text book on mathematics there are 3 exercise A, B, C consisting of 12, 18 and 9 questions respectively. In how many ways can three questions be selected choosing one from each exercise ? In a school, there are four sections of 40 PERMUTATION AND COMBINATION Q.26 Q.27 Q.28 Q.29 Q.30 Q.31 How many 3-digits numbers are three with no digit repeated ? Q.43 How many 4-digit numbers are there with distinct digits? Q.44 (i) How many 2-digit numbers are there ? (ii) How many 3-digit numbers are there ? (iii) How many 4-digit numbers are there ? Q.45 There are 6 books on physics and 5 books on chemistry in a bookshop. In how many ways can a student purchase either a book on physics or a book on chemistry From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done Evaluate: (i) 10P4 (ii) P3 Q.33 (i) If 2nP3 = 100 × nP2, find n. (ii) If 16 nP3 = 13n + 1P3, find n. (iii) If nP5 = 20 × nP3, find n. Q.35 (i) Pr = 13 × 20 P3. 1 n (iii) If 2n (iv) If 2n + 1 n the number of all possible arrangements. Q.46 Seven students are contesting the election for the presidentship of the students’ union. In how many ways can their names be listed on the ballot papers Q.47 Find the number of words formed (may be word ‘EQUATION’, using each letter exactly once. Q.48 : the letters of the word ‘NUMBERS’. Q.49 1 P n = 3 : 5, find n. Q.37 If 15 Q.38 Prove that nPn = 2 nPn  2. Q.39 Find the number of permutations of 7 objects taken 3 at a time. won ? Q.50 many ways can this be done? Q.51 Four toys shaped as the letters B, V, R, S, one of each, were purchased from a plasticware Pr1, find r. shop. How many ordered pairs of letters, to be used as initials, can be formed from them? Pr  2 = 3 : 4, find r. 16 Q.52 If there are 6 periods in each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period ? Q.53 In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are In how many ways can 10 people line up at a ticket window of a cinema hall? In how many ways can 5 children stand in a queue? There are 3 different rings to be worn in 4 fingers with at most one in each finger. In how 12 In how many ways can 5 persons occupy 3 vacant seats ? Ten students are participating in a race. In how many ways can the first three prizes be Pn  1 = 22 : 7, find n. 2n Find the numbers of different 4-letter words (may be meaningless) that can be formed for 2n + 1 11 Q.42 on the same subject are different. Determine P3 = 5 : 12, find n. Pn  1 : Pr  1 : Raju wants to arrange 3 economics, 2 history and 4 language books on a shelf. If the books n+1 If Q.41 mathematics be arranged in a shelf? Pr1, find r. Q.36 Q.40 Pr = P3 : 1 P each in chemistry, physics, biology and 20 If nP4 : nP5 = 1 : 2, find n. (ii) If In how many ways can 4 different books, one 10 If 5Pr = 2 × 6Pr1, find r. (ii) If from 6 wells, if no well remains unused ? (iii) 9P9 62 Prove that 9P3 + 39P2 = (i) In how many ways can 6 women draw water meaningless) by using all the letters of the Q.32 Q.34 Page # 7 to stands in a row behind them? Q.54 How many words can be formed from the letters of the word ‘SUNDAY’? PERMUTATION AND COMBINATION Q.55 Q.56 Q.57 Q.58 Q.59 Q.60 Q.61 Q.62 Q.63 Q.64 Q.65 How many words beginning with C and ending with Y can be formed by using the letters of the word ‘COURTESY’ ? Page # 8 Q.66 seven teachers should be in the first row and all the twenty students should be in the second row. If the two corners of the second row are Find the number of permutations of the letters of the word ‘ENGLISH’. How many of these begin with E and end with I? In how many ways can the letters of the word ‘HEXAGON’ be permuted ? In how many words will the vowel be together ? How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places? reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible? Q.67 that two specified things occur together. Q.68 How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4, 5, 6, 7, 8, when no digit is repeated in any such number ? Q.69 In an examination 10 candidates have to appear. 4 candidates are to appear in mathematics and the rest in different subjects, In how many ways can they be seated in a row, if candidates appearing in mathematics are not to sit together? Q.70 In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together? A child has plastic toys bearing the digits 2, 2 and 5. How many three digits numbers can it make, using them? In how many ways can 6 books on economics, 5 on history and 3 on Hindi be arranged on a shelf so that the books on each subject are always together? Q.71 Find the number of ways in which 5 ladies and 5 gentlemen may be seated in a row so that no two ladies are together. Q.72 Find the number of ways in which m boys and n girls may be arranged in a row so that no two of the girls are together, it being given that m > n. Q.73 In how many ways can 5 children be arranged in a line such that (i) two of them, Ram and Shyam, are always together; (ii) two of them, Ram and Shyam, are never together ? Find a formula for the number of permutations of n different things taken r at a time such In how many words can the letters of the word ‘FAILURE’ be arranged so that the consonants may occupy only odd positions? How many permutations can be formed by the letters of the word ‘VOWELS’, when (i) there is no restriction on letters; (ii)each word begins with E; (iii) each word begin with O and ends with L; (iv) all vowels come together; (v) all consonant come together? When a group photograph is taken, all the There are five round stickers, 3 of them are red the other 2 green. It is desired to make a design by passing them in a row. How many such designs are possible? There are three blue balls, four red balls and five green balls. In how many ways can they be arranged in a row ? How many arrangements can be made out of the letters of the word : (i) Q.74 INDIA (ii) ENGINEERING (iii) COMMERCE (iv) CHANDIGARH (v) INTERMEDIATE (vi) EXAMINATION How many permutations can be made out of the word ‘SERIES’ ? Q.75 How many numbers can be formed with the help of the digits 5, 4, 3, 5, 3? PERMUTATION AND COMBINATION Q.76 Q.77 How many 7-digit numbers can be formed using Page # 9 Q.90 the digits 1, 2, 0, 2, 4, 2 and 4? (i) How many numbers can be formed with the (iv) always occupy the odd place ? How many different signals can be made from 4 red, 2 white and 3 green flags by arranging How many 3-digits numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit In how many ways can 4 letters be posted in 3 letter boxes ? Q.81 In how many ways can 5 prize be distributed among 4 students; when each student may receive any number of prizes? Q.82 How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when A boy has 6 pockets. In how many ways can he put 5 marbles in his pocket? Q.84 In how many ways can 6 persons be arranges Q.85 (i) a line, (ii) a circle ? There are 5 men and 5 ladies to dine at a round table. In how many ways can they seat themselves so that no two no two ladies are together ? Q.86 In how many ways can 11 members of committee sit at a round table so that the secretary and the joint secretary are always the neighbours of the president? Q.87 In how many ways can 8 persons be seated at a round table so that all shall not have the same neighbours in any two arrangements ? Q.88 In how many different ways can 20 different pearls be arranged to form a necklace ? Q.89 (v) Cn n + 1 (i) If nC7 = nC5, find n. (vi)  r 1 5 Cr (ii) If nC14 = nC16, find nC28. (iii) If nC16 = nC14, find nC27. Q.93 (i) If Cr = 20 (ii) If Cr = 18 Cr+6, find r. 20 Cr+2, find rC5. 18 Q.94 If nPr = 1680 and nCr = 70, find n and r. Q.95 If 2n Q.96 If 15 Q.97 If nCr  1 = nC3r, find r. Q.98 If C3 : nC3 11 : 1, find n. Cr : n+1 Cr : n Q.99 Cr  1 = 11 : 5, find r. 15 Cr+1 : nCr = 11 : 6 and n1 Cr1 = 6 : 3, find n and r. Let r and n be positive integers such that 1  r  n. Prove that : n in C71 71 Q.92 a digit may be repeated any number of times? Q.83 (iii) 50C47 Verify that 9C4 + 8C3 = 9C4. may be repeated any number of times Q.80 (ii) 12C9 Q.91 all of them vertically on a flagstaff? Q.79 C3 15 6 digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits Q.78 Evaluate: n 1 Cr n  Cr 1 r Q.100 From a class of 32 students, 4 are to be chosen for a competition. In how many ways can this be done? Q.101 In how many ways can 5 sportsmen be selected from a group of 10? Q.102 In how many ways can a student choose 5 courses out of 9 courses if two courses are compulsory for every student? Q.103 If there are 12 persons in a party and if every two of them shakes hands with each other, how many handshakes happen in the party? Q.104 In how many ways can a cricket team be In how many different ways can a garland of selected from 17 players in which 5 players 16 different flowers be made? can bowl? Each cricket team must include 2 bowlers? PERMUTATION AND COMBINATION Page # 10 Q.105 A question paper has two parts, part ‘A’ and Q.114 From a class of 14 boys and 10 girls, 10 students part ‘B’ each containing 10 questions. If the are to be chosen for a competition, at least students has to choose 8 from part ‘A’ and 5 including 4 boys and 4 girls. The 2 girls who from part ‘B’, in how many ways can he choose won the prizes last year should be included. In the questions ? how many ways can the selection be made ? Q.106 A sports team of 11 students is to be constituted, choosing at least 5 from Class Xl and at least 5 from Class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted ? Q.107 From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the ten be chosen ? Q.108 Out of 6 teachers and 8 students, a, committee of 11 is to be formed. In how many ways can his be done, if the committee contains (i) exactly 4 teachers, (ii) at least 4 teachers Q.109 A cricket team of 11 players is to be selected from 16 players including 5 bowlers and 2 wicketkeepers. In how many ways can a team be selected so as to consist of exactly 3 bowlers and 1 wicketkeeper ? Q.110 Find the numbers of diagonals of (i) a hexagon (ii) a polygon of 16 sides. Q.111 (i) Q.115 In a village, there are 87 families of which 52 families have at most 2 children. In a rural development programme, 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made ? Q.116 A committee of 5 is to be selected from among 6 boys and 5 girls. Determine the number of ways of selecting the committee if it is to consist of at least 1 boy and 1 girl. Q.117 A bag contains 4 red, 3 white and 2 blue marbles. 3 marbles are drawn at random. Determine the numbers of ways of selecting at least 1 white marble in this selection. Q.118 A boy has 3 library tickets and books of his interest in the library. Of these 8, he does not want to borrow Chemistry part II unless Chemistry part I is also borrowed. In how many ways can he choose the three books to be borrowed ? Q.119 In how many ways can 5 white balls and 4 black balls be arranged in a row so that no two black balls are together ? How many straight line can be obtained by joining 12 points, points, 5 of which are Q.120 How many different words, each containing 2 collinear ? vowels and 3 consonants, can be formed with (ii) How many triangles can be obtained by 5 vowels and 17 consonants? joining 12 point, five of which, are collinear ? Q.112 How many (i) lines (ii) triangles can be drawn through n points on a circle ? Q.113 A polygon has 44 diagonals. Find the number of its sides. Q.121 A man has 6 friends. In how many ways can he invite one or more to a party ? Q.122 There are 5 equations in a question paper. In how many ways can a boy solve one more questions ? PERMUTATION AND COMBINATION Page # 11 ANSWER KEY 2. 40320 (i) 504 (ii) 29760 (iii) 1310 6. 455 7. (i) 3 (ii) 49 9. n = 5 10. 90 11. 6 12. 1944 13. 2560000 14. 1050 15. 378 16. 6 17. 9 18. 720 19. 8000 20. 14 21. 900000 22. 56 = 15625 23. 60 24. 36 25. 60 26. 648 27. 4536 28. (i) 90 (ii) 900 (iii) 9000 29. 11 30. 1260 31. (i) 5040 (ii) 226920 (iii) 362880 33. (i) 13 (ii) 15 (iii) 8 34. (i) 3 (ii) 8 35. (i) 6 (ii) 8 (iii) 10 (iv) 4 36. 9 37. 14 39. 210 40. 60 41. 10 ! 42. 120 43. 720 44. 24 45. 9! 46. 5040 47. 40320 48. 840 49. 720 50. 24 51. 12 52. 3600 53. 86400 54. 720 55. 720 56. 5040, 120 57. 5040, 720 58. 576 59. 576 60. (i) 720 (ii) 120 (iii) 24 61. 480 62. 3110400 63. 86400 64. 66. (18!)× (6!)×2 67. 2 68. 12 69. 6! × 7P4 = 604800 72. 27720 74. n–1 Pr 73. (i) 60 (ii) 3. (i) 12! (ii) 35  5! 4! 1. 5. None is true (iv) 240 m! (m  1)! (m  1  n)! 65. (i) 48 (v) 144 (ii) 72 70. 3 71. 10 277200 (iii) 5040 (iv) 907200 (v) 19958400 (vi) 4989600 (i) 180, (ii) 12 75. 30 76. 360 77. 18 78. 1260 81. 45 = 1024 82. 215 83. 65 84. (i) 720 (ii) 120 87. 2520 88. 92. (i) 12 (ii) 435 (iii) 4060 96. 8 97. 104. 2200 93. (i) 7, (ii) 56 1 (n + 1) 98. n = 10,r = 5100. 35960 4 106. 2 × 108. (i) 120 (ii) 344 109. 722 115. C18 × 52 121. 63 C19 × 52 C1 + 35 110. (i) 9, 122. 31 C20116. 455 52 C5 × 20  1) (ii) (1/6) n(n  1)(n  2) C2 + 35 80. 81 85. 2880 86. 80640 1 1 × (19 !) 89. × (15 !) 90. (i) 455 (ii) 220 (iii) 19600 (iv) 1 (v) n + 1 (vi) 31 2 2 105. 11340 112. (i) (1/2) n(n 79. 100 C6 20 (ii) 104 94. n = 8, r = 4 95. 6 101. 252 103.66 102. 35 107. 817190 111. (i) 57, (ii) 113. 11 114. 266266 117. 64 118. 41 119. 20 210 120.816000