Title 7-permutations--combinations.pdf ✅

Permutations & Combinations 1. In how many ways 6 letters can be placed in 6 envelopes such that no letter is placed in its corresponding envelope. (A) 265 (B) 275 (C) 255 (D) None of these 2. Find the sum of all four digits numbers that can be formed using the digits 0,1,2,3,4; no digits being repeated in any number. (A) 259980 (B) 249980 (C) 269960 (D) None of these 3. In a plane there are 37 straight line, of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through both points A and B, and no two are parallel. Find the number of points of intersection of the straight lines. (A) 533 (B) 536 (C) 535 (D) 530 4. There are p points in a plane, no three of which are in the same straight line with the exception of q , which are all in the same straight line. Straight lines which can be formed by joining them : (A) pC2 − qC2 (B) pC2 − qC2 + 1 (C) pC2 + qC2 (D) pC2 + qC2 + 1 5. 10 different toys are to be distributed among 10 children. Total number of ways of distributing these toys so that exactly 2 children do not get any toy, is equal to : (A) (10!) 2 ( 1 3!2!7! + 1 (2!) 56! ) (B) (10!) 2 ( 1 3!2!7! + 1 (2!) 46! ) (C) (10!) 2 ( 1 3!7! + 1 (2!) 56! ) (D) (10!) 2 ( 1 3!7! + 1 (2!) 46! ) 6. In how many ways we can divide 52 playing cards among 4 players equally ? (A) 52! (13!) 4 × 1 4 (B) 52! (13!) 4 (C) 52!4 (13!) 4 (D) None of these 7. In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two women are adjacent? (A) 144 (B) 24 (C) 72 (D) 288 8. In how many ways can they be seated if two particular delegates are always together ? (A) 18 ! (B) 2 × 18 ! (C) 17 ! (D) 2 × 17 ! 9. In how many ways can they be seated if two particular delegates are never together ? (A) 17 × 18 ! (B) 18 × 18 ! (C) 15 × 18 ! (D) 16 × 18 ! 10. If the number of ways of selecting k coupons one by one out of an unlimited number of coupons bearing the letters A, T, M so that they cannot be used to spell the word MAT is 93 , then k equals to : (A) 3 (B) 5 (C) 7 (D) 8 11. Total number of positive integral solution of 15 < x1 + x2 + x3 ≤ 20, is equal to : (A) 1125 (B) 1150 (C) 1245 (D) 685 12. The number of permutations of the letters of the HINDUSTAN such that neither the pattern 'HIN' nor 'DUS' nor 'TAN' appears, are : (A) 166674 (B) 169194
(C) 166680 (D) 181434 13. If 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide the 3 n things in 3 equal groups is : (A) k × 3 ! (B) k 3! (C) (3!) k (D) 3k 14. Let A be the set of 4-digit numbers a1a2a3a4 where a1 > a2 > a3 > a2 then n( A) is equal to : (A) 126 (B) 84 (C) 210 (D) 120 15. A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are 5 places in row in his showcase. The number of different ways of displaying the three varieties of perfumes in the showcase is: (A) 6 (B) 50 (C) 150 (D) 90 16. The number of subsets of the set A = {a1, a2, ... , an } which contain even number of elements is : (A) 2 n−1 (B) 2 n − 1 (C) 2 n−2 (D) 2 n 17. How many numbers are there between 100 and 1000 in which all the digits are distinct ? (A) 648 (B) 729 (C) 576 (D) 810 18. Six X have to be placed in the squares of figure such that each row contains at least one X. The number of ways in which this can be done is : (A) 25 (B) 26 (C) 27 (D) 30 19. If all the letters of the word 'AGAIN' be arranged as in a dictionary, what is the fiftieth word? (A) NAAIG (B) NAAGI (C) NAGAI (D) NAGIA 20. The range of the function 7−xPx−3 is: (A) {1,2,3,4} (B) {1,2,3,4,5,6} (C) {1,2,3} (D) {1,2,3,4,5} 21. The results of 21 football matches (win, lose or draw) are to be predicted. The number of forecasts that contain exactly 18 correct results is : (A) 21C32 18 (B) 21C182 3 (C) 3 21 − 2 18 (D) 21C33 21 − 2 18 22. Letters of the word INDIALOIL are arranged in all possible ways. The number of permutations in which A,I, O occur only at odd places, is : (A) 720 (B) 360 (C) 240 (D) 120 23. At an election there are five candidates and three members are to be elected, and a voter may vote for any number of candidates not greater than the number to be elected. The number of ways in which the person can vote is : (A) 25 (B) 30 (C) 35 (D) 2 5 − 2 3 24. A five digit number divisible by 3 is to be formed using the digits 0,1,2,3,4 and 5 without repetition. The total number of ways, in which this can be done is : (A) 240 (B) 3125 (C) 600 (D) 216 25. The number of ways in which two teams A and B of 11 players each can be made up from 22 players so that two particular players are on the opposite sides is : (A) 369512 (B) 184755
(C) 184756 (D) 369514 26. A library has a copies of one book, b copies of each of two books, c copies of each of three books and single copies of d books. The total number of ways in which these books can be distributed is : (A) (a+b+c+d)! a!b!c! (B) (a+2b+3c+d)! a!(b!) 2(c!) 3 (C) (a+2b+3c+d)! a!b!c! (D) None of these 27. Given that n is odd, the number of ways in which three numbers in AP can be selected from 1,2,3, ... , n is : (A) (n−1) 2 2 (B) (n+1) 2 4 (C) (n+1) 2 2 (D) (n−1) 2 4 28. 20 persons are invited for a party. In how many different ways can they and the host be seated at circular table, if the two particular persons are to be seated on either side of the host? (A) (B) (C) (D) None of these 29. Assuming that no two consecutive digits are same, the number of n-digit numbers is : (A) ⌊n (B) ⌊9 (C) 9 n (D) n 9 30. In how many ways can 21 identical English and 19 identical Hindi books be placed in a row so that no two Hindi books are together? (A) 1540 (B) 1450 (C) 1504 (D) 1405 31. If 2n+1Pn−1: 2n−1Pn = 3: 5, then n equals: (A) 5 (B) 4 (C) 3 (D) 2 32. The greatest number of points of intersection of n circles and m straight lines is : (A) 2mn + mC2 (B) 1/2m(m − 1) + n(2m + n − 1) (C) mC2 + 2( nC2 ) (D) None of these 33. There are three pigeon holes marked M, P, C. The number of ways in which we can put 12 letters so that 6 of them are in M, 4 are in P and 2 are in C is : (A) 2520 (B) 13860 (C) 12530 (D) 25220 34. Three straight lines 11, 12 and 13 are parallel and lie in the same plane. Five points are taken on each of 11, 12 and 13. The maximum number of triangles which can be obtained with vertices at these points, is : (A) 425 (B) 405 (C) 415 (D) 505 35. There are 15 points in a plane, no three of which are in a straight line except 4 , all of which are in a straight line. The number of triangles that can be formed by using these 15 points is: (A) 404 (B) 415 (C) 451 (D) 490 36. If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is R and the fourth letter is E, then the total number of all such words is : (A) 11! (2!) 3 (B) 110 (C) 56 (D) 59 37. (A) n 2 − 3n − 108 = 0 (B) n 2 + 5n − 84 = 0 (C) n 2 + 2n − 80 = 0 (D) n 2 + n − 110 = 0 38. The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is : (A) 1120 (B) 1240
(C) 1880 (D) 1960 39. Let A = {x1, x2, ... ... , x7 } and B = {y1, y2, y3 } be two sets containing seven and three distinct elements respectively. Then the total number of functions f: A → B that are onto, if there exists exactly three elements x in A such that f(x) = y2, is equal to : (A) 14 ⋅ 7C2 (B) 16 ⋅ 7C3 (C) 12 ⋅ 7C2 (D) 14 ⋅ 7C3 40. The sum of the digits in the unit's place of all the 4-digits numbers formed by using the numbers 3,4,5 and 6 , without repetition is: (A) 432 (B) 108 (C) 36 (D) 18 41. An eight digit number divisible by 9 is to formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is : (A) 72(7!) (B) 18(7!) (C) 40(7!) (D) 36(7!) 42. Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval: (A) [8,9] (B) [10,12) (C) (11,13] (D) (14,17) 43. The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less then 2 marks to any question, is : (A) 30C7 (B) 21C7 (C) 21C7 (D) 30C8 44. All possible three digits even numbers which can be formed with the condition that if 5 is one of the digit then 7 is the next digit is: 45. The number of arrangements of the letters ' abcd ' in which neither a, b nor c, d come together is: 46. The number of three-digit numbers having only two consecutive digits identical is 47. A committee of 5 is to be chosen from a group of 9 people. Number of ways in which it can be formed if two particular persons either serve together or not at all and two other particular persons refuse to serve with each other, is 48. If m denotes the number of 5 -digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure, when the digits are in their ascending order of magnitude then (m − n) has the value 49. A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair is 50. An old man while dialing a 7-digit telephone number remembers that the first four digits consists of one 1 's, one 2's and two 3 's. He also remembers that the fifth digit is either a 4 or 5 while has no memorising of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. Maximum number of distinct trials he has to try to make sure that he dials the correct telephone number, is: 51. Number of three-digit number with atleast one 3 and at least one 2 is 52. In a unique hockey series between India & Pakistan, they decide to play on till a team wins 5 matches. The number of ways in which the series can be won by India, if no match ends in a draw is: 53. Define a 'good word' as a sequence of letters that consists only of the letters A, B and C and in which A never immediately followed by B, B is never immediately followed by C , and C is never immediately followed by A. If the number of n-letter good words are 384, find the value of n. 54. Fifty college teachers are surveyed as to their possession of colour TV, VCR and tape recorder. Of them, 22 own colour TV, 15 own VCR and 14 own tape recorders. Nine of these college teachers own exactly two items out of colour TV, VCR and tape recorders; and, one college teacher owns all three, how many of the 50 college teachers own none of three, colour TV, VCR or tape recorder? 55. Six people are going to sit in a row on a bench. A and B are adjacent, C does not want to sit