Permutations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Objective Sheet - 1
1. How many numbers divisible by 5 and 8. In how many ways can mn letters be
lying between 3000 and 4000 can be posted in n letter-boxes
formed from the digits 1, 2, 3, 4, 5, 6
(a) (mn)n (b) mmn
(repetition is not allowed)
(a) 6 P (b) 5P (c) nmn (d) None
2 2 9. In how many ways can 10 true-false
(c) 4 P (d) 6P questions be replied
2 3 (a) 20 (b) 100
2. The number of ways in which 6 rings can (c) 512 (d) 1024
be worn on the four fingers of one hand is n
10. The value of Pr is equal to
(a) 46 (b) 6C
4 [IIT 1971]
(c) 64 (a) n 1 P r n 1P
(d) None r r 1
(b) n. n 1P n 1P
3. How many numbers can be formed from
the digits 1, 2, 3, 4 when the repetition is r r 1
not allowed
(c) n( n 1 P n 1P )
r r 1
(a) 4 P
(d) n 1 P n 1 P
4
r 1 r
(b) 4 P
3 11. Find the total number of 9 digit numbers
which have all the digits different
(c) 4 P 4 P 4 P
1 2 3 [IIT 1982]
(d) 4 P P P 4 P
4 4 (a) 9 9 ! (b) 9!
1 2 3 4 (c) 10 ! (d) None
4. There are 3 candidates for a post and one 12. Four dice (six faced) are rolled. The
is to be selected by the votes of 7 men. number of possible outcomes in which at
The number of ways in which votes can least one die shows 2 is
be given is (a) 1296 (b) 625
(a) 73 (b) 37 (c) 671 (d) None
7C 13. There are 4 parcels and 5 post-offices. In
(c) (d) None how many different ways the registration
3
5. 4 buses runs between Bhopal and of parcel can be made
Gwalior. If a man goes from Gwalior to (a) 20 (b) 45
Bhopal by a bus and comes back to
Gwalior by another bus, then the total (c) 54 (d) 54 45
possible ways are 14. In how many ways can 5 prizes be
(a) 12 (b) 16 distributed among four students when
(c) 4 (d) 8 every student can take one or more prizes
(a) 1024 (b) 625
6. If n P 20. n P , then n = (c) 120 (d) 600
5 3
(a) 4 (b) 8 15. The product of any r consecutive natural
(c) 6 (d) 7 numbers is always divisible by
7. How many words comprising of any three (a) r ! (b) r2
letters of the word UNIVERSAL can be
formed (c) rn (d) None
(a) 504 (b) 405
(c) 540 (d) 450
SCO – 35, 2nd Floor, Sector 19/D, Chd. 98151-62400, 9988262400 1
Permutations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16. The sum of the digits in the unit place of (c) 7! (d) None
all numbers formed with the help of 3, 4, 26. In how many ways n books can be
5, 6 taken all at a time is arranged in a row so that two specified
(a) 18 (b) 432 books are not together
(c) 108 (d) 144 (a) n ! (n 2)! (b) (n 1)!(n 2)
17. Six identical coins are arranged in a row. (c) n ! 2( n 1) (d) (n 2) n !
The number of ways in which the number 27. Numbers greater than 1000 but not greater
of tails is equal to the number of heads is than 4000 which can be formed with the
(a) 20 (b) 9 digits 0, 1, 2, 3, 4 (repetition of digits is
(c) 120 (d) 40 allowed), are [IIT 1976; AIEEE 2002]
18. The figures 4, 5, 6, 7, 8 are written in (a) 350 (b) 375
every possible order. The number of (c) 450 (d) 576
numbers greater than 56000 is 28. The number of numbers that can be
(a) 72 (b) 96 formed with the help of the digits 1, 2, 3,
(c) 90 (d) 98 4, 3, 2, 1 so that odd digits always
19. The sum of all 4 digit numbers that can be occupy odd places, is
formed by using the digits 2, 4, 6, 8 (a) 24 (b) 18
(repetition of digits not allowed) is (c) 12 (d) 30
(a) 133320 (b) 533280 29. In how many ways can 5 boys and 3 girls
(c) 53328 (d) None sit in a row so that no two girls are
20. There are 5 roads leading to a town from together
a village. The number of different ways in 4 P 5 !
which a villager can go to the town and (a) 5 ! 3 ! (b)
3
return back, is
(a) 25 (b) 20 (c) 6 P 5 ! (d) 5 P 3!
3 3
(c) 10 (d) 5 30. How many numbers less than 1000 can be
21. In how many ways can five examination made from the digits 1, 2, 3, 4, 5, 6
papers be arranged so that physics and (repetition is not allowed)
chemistry papers never come together (a) 156 (b) 160
(a) 31 (b) 48 (c) 150 (d) None
(c) 60 (d) 72 31. How many words can be made from the
22. The number of 3 digit odd numbers, that letters of the word DELHI, if L comes in
can be formed by using the digits 1, 2, 3, the middle in every word
4, 5, 6 when the repetition is allowed, is (a) 12 (b) 24
(a) 60 (b) 108 (c) 60 (d) 6
(c) 36 (d) 30 32. In how many ways 3 letters can be posted
12
23. If P 1320 , then r is equal to in 4 letter-boxes, if all the letters are not
r posted in the same letter-box
(a) 5 (b) 4 (a) 63 (b) 60
(c) 3 (d) 2 (c) 77 (d) 81
24. The numbers of arrangements of the 33. The number of 5 digit telephone numbers
letters of the word SALOON, if the two having at least one of their digits repeated
O's do not come together, is is
(a) 360 (b) 720 (a) 90,000 (b) 100,000
(c) 240 (d) 120 (c) 30,240 (d) 69,760
25. The number of words which can be 34. The number of arrangements of the letters
formed from the letters of the word of the word CALCUTTA
MAXIMUM, if two consonants cannot (a) 2520 (b) 5040
occur together, is (c) 10,080 (d) 40,320
(a) 4! (b) 3! 4!
SCO – 35, 2nd Floor, Sector 19/D, Chd. 98151-62400, 9988262400 2
Permutations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
35. In a circus there are ten cages for (c) 30 (d) 720
accommodating ten animals. Out of these 42. How many numbers greater than 24000
four cages are so small that five out of 10 can be formed by using digits 1, 2, 3, 4, 5
animals cannot enter into them. In how when no digit is repeated
many ways will it be possible to (a) 36 (b) 60
accommodate ten animals in these ten (c) 84 (d) 120
cages [Roorkee 1989] 43. How many numbers greater than hundred
(a) 66400 (b) 86400 and divisible by 5 can be made from the
(c) 96400 (d) None digits 3, 4, 5, 6, if no digit is repeated
36. The letters of the word MODESTY are (a) 6 (b) 12
written in all possible orders and these (c) 24 (d) 30
words are written out as in a dictionary, 44. The number of 7 digit numbers which can
then the rank of the word MODESTY is be formed using the digits 1, 2, 3, 2, 3, 3,
(a) 5040 (b) 720 4 is
(c) 1681 (d) 2520 (a) 420 (b) 840
37. If a denotes the number of permutations (c) 2520 (d) 5040
of x 2 things taken all at a time, b the 45. The number of 4 digit numbers that can
number of permutations of x things taken be formed from the digits 0, 1, 2, 3, 4, 5,
11 at a time and c the number of 6, 7 so that each number contain digit 1 is
permutations of x 11 things taken all at (a) 1225 (b) 1252
a time such that a 182 bc , then the (c) 1522 (d) 480
value of x is 46. The number of 4 digit even numbers that
(a) 15 (b) 12 can be formed using 0, 1, 2, 3, 4, 5, 6
(c) 10 (d) 18 without repetition is
38. The number of ways in which ten (a) 120 (b) 30
candidates A , A , ....... A can be ranked (c) 420 (d) 20
1 2 10 47. Total number of four digit odd numbers
such that A1 is always above A is that can be formed using 0, 1, 2, 3, 5, 7
10
are [AIEEE 2002]
(a) 5! (b) 2(5 !)
(a) 216 (b) 375
1 (c) 400 (d) 720
(c) 10 ! (d) (10 !)
2 48. The number of arrangements of the letters
39. All the letters of the word ‘EAMCET’ are of the word BANANA in which two N’s
arranged in all possible ways. The number do not appear adjacently is
of such arrangements in which two [IIT Screening 2002]
vowels are not adjacent to each other is (a) 40 (b) 60
[DEC 2000] (c) 80 (d) 100
(a) 360 (b) 114 49. The number of ways in which 5 boys and
(c) 72 (d) 54 3 girls can be seated in a row so that each
40. In how many ways can 5 boys and 5 girls girl in between two boys
stand in a row so that no two girls may be (a) 2880 (b) 1880
together (c) 3800 (d) 2800
(a) (5 !)2 (b) 5 ! 4 ! 50. Eleven books consisting of 5
Mathematics, 4 Physics and 2 Chemistry
(c) 5 ! 6 ! (d) 65!
are placed on a shelf. The number of
41. The number of words which can be made possible ways of arranging them on the
out of the letters of the word MOBILE assumption that the books of the same
when consonants always occupy odd subject are all together is
places is (a) 4! 2! (b) 11!
(a) 20 (b) 36 (c) 5! 4! 3! 2! (d) None
SCO – 35, 2nd Floor, Sector 19/D, Chd. 98151-62400, 9988262400 3
Permutations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
51. If the letters of the word SACHIN 5 ! 5 !
arranged in all possible ways and these (c) (d) None
2
words are written out as in dictionary, 4. In how many ways can 12 gentlemen sit
then the word SACHIN appears at serial around a round table so that three
number [AIEEE 2005] specified gentlemen are always together
(a) 603 (b) 602 (a) 9! (b) 10!
(c) 601 (d) 600 (c) 3! 10! (d) 3! 9!
52. Let the eleven letters A, B .....,K denote an 5. In how many ways can 15 members of a
arbitrary permutation of the integers (1, council sit along a circular table, when the
2,.....11), then Secretary is to sit on one side of the
( A 1)( B 2)(C 3).....( K 11) Chairman and the Deputy Secretary on
(a) Necessarily zero the other side
(b) Always odd (a) 2 12! (b) 24
(c) Always even (c) 2 15! (d) None
(d) None of these 6. 20 persons are invited for a party. In how
53. 4 Note of Rs. 100 and 5 note in which many different ways can they and the host
first of Rs. 1, second of Rs. 2, Third of be seated at a circular table, if the two
Rs. 5, fourth of Rs. 20 and fifth one of Rs. particular persons are to be seated on
50 distributed in 3 children such that each either side of the host [IIT 1977]
child receive at least one note of Rs. 100. (a) 20! (b) 2. 18!
The total number of ways of distribution (c) 18! (d) None
(a) 3 53 (b) 5 35 7. The number of ways in which 5 beads of
different colures form a necklace is
(c) 36 (d) None (a) 12 (b) 24
Circular permutations (c) 120 (d) 60
1. If eleven members of a committee sit at a 8. The number of ways in which 5 male and
round table so that the President and 2 female members of a committee can be
Secretary always sit together, then the seated around a round table so that the
number of arrangements is two female are not seated together is
(a) 10 ! 2 (b) 10! [Roorkee 1999]
(c) 9! 2 (d) None (a) 480 (b) 600
(c) 720 (d) 840
2. In how many ways can 5 keys be put in a
9. In how many ways 7 men and 7 women
ring
can be seated around a round table such
1 1 that no two women can sit together
(a) 4! (b) 5!
2 2
(c) 4! (d) 5! (a) (7!)2 (b) 7! 6!
3. In how many ways can 5 boys and 5 girls (c) (6!)2 (d) 7!
sit in a circle so that no two boys sit 10. The number of ways in which 6 men and
together [IIT 1975] 5 women can dine at a round table if no
(a) 5! 5! (b) 4! 5! two women are to sit together is given by
(a) 6! × 5! (b) 30
(c) 5! × 4! (d) 7! × 5!
SCO – 35, 2nd Floor, Sector 19/D, Chd. 98151-62400, 9988262400 4
Permutations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Answers
1 (c) 2 (a) 3 (d) 4 (b) 5 (a)
6 (b) 7 (a) 8 (c) 9 (d) 10 (a)
11 (a) 12 (c) 13 (c) 14 (a) 15 (a)
16 (c) 17 (a) 18 (c) 19 (a) 20 (a)
21 (d) 22 (b) 23 (c) 24 (c) 25 (a)
26 (b) 27 (b) 28 (b) 29 (c) 30 (a)
31 (b) 32 (b) 33 (d) 34 (b) 35 (b)
36 (c) 37 (b) 38 (d) 39 (c) 40 (c)
41 (b) 42 (c) 43 (b) 44 (a) 45 (d)
46 (c) 47 (d) 48 (a) 49 (a) 50 (c)
51 (c) 52 (c) 53 (c)
Circular permutations
1 (c) 2 (a) 3 (b) 4 (d) 5 (a)
6 (b) 7 (a) 8 (a) 9 (b) 10 (a)
SCO – 35, 2nd Floor, Sector 19/D, Chd. 98151-62400, 9988262400 5
