MULTIPLE CHOICE QUESTIONS PERMUTATION AND COMBINATION

1) Value of 2ⁿ{1.3.5.7....(2n -1)} is

a) ²ⁿPₙ b) ⁿPₙ c) ²ⁿCₙ d) none

2) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5, find n

a) 4 b) 4 c) 5 d) 6

3) ⁿC₃ =120 then n is

a) 10 b) 12 c) 14 d) none

4) If ⁿPₓ=336 and ⁿCₓ =56 find n and x.  

a) 3,8 b) 8,3 c) 6,5 d) 5,6

5) ⁿCₓ + ⁿCₓ₋₁ is

a) ⁿ⁺¹Cₓ b) ⁿCₓ c) ⁿ⁺¹Pₓ  d) ⁿPₓ 

6)⁴₂⁴₁⁵₂¹⁸ₓ¹⁸ₓ₊₂ˣ₅

6) If n and x are positive integers and x≤ n,  then which of the following is correct ?

a) ⁿPₓ = ⁿ⁻¹Pₓ₋₁  b) ⁿPₓ = x. ⁿ⁻¹Pₓ₋₁

c) ⁿPₓ = n. ⁿ⁻¹Oₓ₋₁ d) none

7) How many numbers of 5 digits can be formed from the number 2,0, 4, 3, 8, when repetition of digits is not allowed ?

a) 96 b) 120 c) 144 d) 14

8) Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points ?

a) 26  b) 28 c) 27 d) 25

9) The number of ways in which n different things can be put in 3 different boxes is

a) n³ b) 3. n! c) 3n² d) 3ⁿ

10) The number of ways in which n white balls and n red balls can be arranged in a row so that the balls of the same colour are not consecutive, is

a) 2(n!)² b) (n!)² c) 2(n!) d) (2n)!

11) If n= ᵐC₂, then the value of ⁿC₂ will be

a) ᵐ⁺¹C₄ b) ᵐ⁺²C₄ c) 3 . ᵐ⁺¹C₄ d) 3. ᵐ⁺²C₄

12) ²ⁿPₙ is equal to

a) (n+1)! . ²ⁿCₙ. b) n! . ²ⁿCₙ

c) n! . ²ⁿ⁺¹Cₙ      d) ²ⁿ⁺¹Cₙ₊₁

13) The letter of the word TRIANGLE are arranged in a row in all possible ways. How many of them begin with A and end with N?

a) 60 b) 120 c) 720 d) 1680

14) If n is a  natural number, then the value of (2n)!/n! is

a) 2.4.6...2n b) 2ⁿ c) 2²ⁿ  d) 1.3.5...(2n -1).2ⁿ

15) If ²⁰Cₓ = ²⁰Cₓ₊₄, then the value of x is.  

a) 8 b) 7 c) 6 d) 5

16) The number of ways in which n different things can be put into two boxes so that none of the boxes is empty, is

a) 2ⁿ -2 b) n²-2 c) 2ⁿ -1 d) n²- 1

17) how many 7 digit numbers can be formed using the digits 1,2 find 3 onoso that the digit 2 occurs twice in each number?

a) ⁷P₂ . 2⁵ b) ⁷P₂. 5² c) ⁷C₂. 2⁵ d) ⁷C₂. 5²

18) If ¹⁵Cₓ = ¹⁵Cₓ₊₁ , then the value of ˣPₓ₋₃ is

a) 31 b) 120 c) 210 d) 840

19) Which one of the following is true?

a) ⁿCₓ + ⁿCₓ₋₁= ⁿCₓ₊₁

b) ⁿCₓ + ⁿ⁻¹Cₓ₋₁ = ⁿ⁺¹Cₓ

c) ⁿ⁻¹Cₓ + ⁿ⁻¹Cₓ₋₁ = ⁿₓ  d) none

20) The number of ways in which the letter of the word ARRANGE can be arranged such that both R do not come together is

a) 360 b) 900 c) 1260 d) 1620. 

21) 1/8! + 1/9! = x/10! , then the value of x is

a) 100 b) 10 c) 81 d) 64

22) How many one digit or more than one with the digit numbers can be formed with the digits 0, 1, 2, 3, 4, 5, none of the digits being repeated in any of the numbers so formed?

a) 1560 b) 1400 c) 1540 d) 1630

23) How many triangles can be formed from the vertices of an octagon so that no side of the octagon is a side of the triangle?

a) 48 b) 16 c) 24 d) 56

24) The total number of ways in which m positive and negative n(< m+1) negative signs can be arranged in a line such that no two negative signs occur together is

a) ᵐ⁺¹Cₙ b) ᵐ⁺¹Pₙ c) ⁿ⁺¹Cₘ d) ⁿ⁺¹Pₘ

25) A college library has m copies of each of n different books. The total number of ways in which one or more books can be selected from the library is

a) (m+1)ⁿ b) (m+1)ⁿ -1 c) (m+1)ⁿ - n d) (m+1)ⁿ - m

26) How many odd number of six significant digits can be formed with the digits 0, 1, 2, 5, 6, 7 when no digit is repeated ?

a) 120 b) 96 c) 360 d) 288

27) If (n+2)!= 182 . n!,  then the value of n is

a) 12  b) 13 c) 14 d)  15

28) if n is a natural number, in which of the following is true ?

a) ⁿPₙ = 12 x ⁿPₙ₋₄ b) ⁿPₙ = 6 x ⁿPₙ₋₃

c) ⁿPₙ = 12 x ⁿPₙ₋₃ d) none 

29) If ⁿC₁₅ = ⁿC₈, then the value of ⁿC₂₁ is

a) 46 b) 506 c) 253 d) 23

30) If x. ⁿ⁻¹Cₓ₋₁ = ⁿ⁻²Cₓ₋₂, then the value of x is

a) (n-1)/(x-1) b) (n-1)/(x-2) c) (x-1)/(n-1) d) (x-2)/(n-2)

31) The number of ways in which 10 different biscuits can be distributed among 6 children is

a) ¹⁰P₆ b) 10⁶ c) 10 . ¹⁰P₆ d) 6¹⁰

32) The number of ways in which 4 numbers can be selected from first 25 natural numbers so that the selected numbers are not consecutive, is

a) 12650 b) 12628 c) 12672 d) 12600

33) If ⁿ⁺¹Cₙ₋₂ + ⁿ⁺¹Cₙ₋₁ =165, the value of n is

a) 7 b) 8 c) 9 d) 10

34) How many number of 6 significant digit can be formed from the seven digit 3, 4, 5, 6, 7, 8, 9 so that the extreme two digits are always even (no digit being represted in any number)?

a) 720 b) 360 c) 144  d) 288

35) If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃ then which of the following is true?

a) n> 4 b) n >5 c) n > 6 d) n > 7

36) The number of different ways in which a man can post 4 letters in 6 letter boxes is

a) ⁶P₄ b) 4⁶ c) 6⁴ d) 6⁴ -1

37) n. ⁿ⁻¹Cₓ₋₁ = x. ⁿCₓ₋₁ , then the value of x is

a) n-r b) n-r - c) r d) n-r+1

38) The sum of the numbers which can formed the digit 2, 3, 4, 5 only once is 

a) 93324  b) 15554  c) 26664 d) none

39) The value of ⁿCₙ + ⁿ⁺¹Cₙ + ⁿ⁺²Cₙ is

a) ⁿ⁺⁴Cₙ b) ⁿ⁺³Cₙ c) ⁿ⁺²Cₙ₊₁ d) ⁿ⁺³Cₙ₊₁

40) If ¹²Cₓ = ¹²C₂ₓ₋₃ (x≠3), then the value of x is

a) 4 b) 5 c) 6 d) 7 

41) If ¹¹P₆ + 6. ¹¹P₅ = ¹²Pₐ , then the value of a is

a) 5 b) 6 c) 7 d) 4

42) If the number of diagonals of a polygon is 54, then the number of its sides will be

a) 14  b) 13 c) 11  d) 12 

43) There are 15 points in a plane, of which no three are collinear except 8 points. Then the number of triangles formed by the points will be

a) 399  b) 400 c) 455 d) 456

44) ⁿ⁻¹C₇ + ⁿ⁻¹C₆ > ⁿC₆, then the value of n is

a) >13  b) ≥ 13  c) > 12 d) ≥ 12

45) There are 25 railway station on a certain railway line then. Then the number of different kinds of tickets of class II must be painted in order that a passenger may go from any one station to any other, is 

a) ²⁵C₂ b) ²⁵P₂ c) 2 x ²⁵P₂ d) none

46) There are n balls in each of two bags. A man draws equal number of balls from each bag. The number of different ways in which he can draw atleast one ball from each of the bags is

a) ²ⁿCₙ b) (²ⁿCₙ)² c) ²ⁿCₙ - 1 d) ²ⁿCₙ + 1

47) How many number greater than 5000000 can be formed with 0, 3, 4, 4, 5, 6, 6 ?

a) 360 b) 720 c) 420 d) 540 

48) l₁ , l₂ , l₃ are three parallel lines and they are on the same plane. If m, n  and p points are taken on the lines l₁ , l₂ , l₃ respectively, then the number of triangles formed by taking these points as vertices of a triangle is

a) ᵐ⁺ⁿ⁺ᵖC₃ b) ᵐC₃ + ⁿC₃ + ᵖC₃ c) ᵐC₃ + ⁿC₃+ ᵖC₃ - 3 d) none

49) The sum of all 2-digit odd numbers is

a) 2475 b) 2530 c) 4905 d) 5049

50) The value of ⁿᵣ₌₁∑ ⁿPᵣ/r! is

a) 2ⁿ b) 2ⁿ -1 c) 2ⁿ+1 d) n. 2ⁿ⁻¹

51) Nine questions are given ; each question has an alternative. The number ways in which a student can answer one or more questions , is

a) 2⁹ -1 b) 3⁹ c) 2⁹ d) 3⁹ -1

52) How many numbers of 6 significant digits divisible by 4 can be formed using the digits 0, 1, 2, 3, 4, 5, no digit being repeated in any number ?

a) 144 b) 126 c) 72 d) 108 

53) In how many different ways three numbers can be selected from the first 20 natural numbers so that the selected three numbers are not consecutive ?

a) 1124 b) 1122 c) 1120  d) 1118

54) If ³⁵Cₙ₊₇ = ³⁵C₄ₙ₋₂ , then find the value of of n

a) 5 b) 7 c) 4 d) 6 or 3

55) ⁿPₓ =504 and ⁿCₓ =84, then the value of n is

a) 8 b) 7 c) 6 d) 9

56) If ²⁰Oₓ₊₃ : ¹⁸Pₓ = 3800:1, then find the value of x is

a) 9 b) 8 c) 12 d) 15

57) Everybody in a room shakes hands with everybody else. The total number of handshakes is 105. Then the total number of person in the room is

a) 15 b) 10 c) 12 d) 18 

58) The number of diagonals of a polygon having 20 sides is

a)  140 b) 150 c) 170 d) 160 

59) The least value of r for which ¹²Cᵣ₋₁ > ¹²Cᵣ is

a) 10 b) 9 c) 11 d) 8 

60) The number of 6 digit numbers which can be formed using each of the digits 5, 6, 7, 8 at least once, is

a) 1280  b) 560 c) 840 d) 1560

61) How many even numbers of 6 significant digits can be formed with the digit 9, 8, 7, 6, 5, 0 when no digit is repeated.

a) 288 b) 296  c) 312 d) 320

62)  7 speakers A, B, C, D, E, F and G will speak at a meeting. In how many ways can they take their turns if A always speaks before B?

a) 2430 b) 2520 c) 3240 d) 2560

63) The total number of factors of 38808 (excluding 1 and 38808) is

a) 72 b) 71 c) 69 d) 70 

64) In a chess championship every player played one game with each other; if the number of games played by 66, then the number of players participating in the championship?

a) 12 b) 11 c) 10 d) 13 

65) A committee of 5 is to be formed from 6 gentlemen and seven ladies . The number of committees with at least one gentleman and a majority of ladies is

a) 210 b) 525 c) 735 d) 756 

66) The number of different ways in which a pack of 52 cards can be divided equally among 4 players, is

a) 4 x ⁵²C₁₃ b) ⁵²C₄ c) ⁵²C₄ x 1/4! d) (52)!/(13!)⁴

67) A parallelogram is cut by two sets of n lines parallel to its sides. The number of parallelogram thus formed is

a) n²/4 (n+1)² b) 1/4 (n+2)²(n+1)² 

c) 1/4 n²(n+1)²   d) 1/4 (n+1)²(n-1)² 

68) A box contain 4 white, 3 red and 2 black balls. The number of ways in which 3 balls can be drawn from the box, so that at least one of the three drawn balls is red, is

a) 64 b) 63 c) 72 d) 80

69) The number of ways in which 5 boys and 5 girls can be arranged in a line so that boys and girls occupy consecutive positions is

a) (5!)² b) 2. (5!)² c) 2. 5! ⁶P₅ d) (10!)

70) There are 4, 5 and 6 points on the side BC, CA and AB respectively of the triangle ABC. The number of triangles formed by taking these given points as vertices of a triangle, is

a) 421 b) 425 c) 455 d) 411

71) A gentleman gives a dinner party to 6 guests to be selected from his 10 friends . The number of ways of the forming the party of 6, given that 2 of the friends will not attend the party together, is

a) 210 b) 140 c) 70 d) none

72) A library has m copies of each of n different books. Then the total number of ways in which one or more of the books can be selected, is

a) (n+1)ᵐ b) (m+1)ⁿ c) (m+1)ⁿ -1 d) (n+1)ᵐ -1

73) The number of ways in which one or more fruits can be selected from 4 bananas , apples and 9 oranges is

a) 251 b) 400 c) 252 d) 399 

74) 10 different letters of an alphabet are given and words with 5 letters are form formed these given letters. Then the number of words which have at least one letter repeated is 

a) 50360  b) 69760 c) 40240 d) none 

75) The sum of the digits of hundred's place of all 4-digit numbers which can be formed with the digits 3,4,5,6 (where no digit repeated) is

a) 108 b) 1080 c) 48 d) 216

76) Let, x denote the number of words which can be formed from the letters of the word IMPLICIT so that letters M and P do not come together and y denote the number of words which begin with M and end with P; then the value of x/y is

a) 42  b) 35 c) 48 d) 28 

77) The number of different signals which can be made by selecting one or more of 6 flags of different colours , is

a)  1960 b) 1956  c) 216 d) 1950 

78) The number of ways in which 3 letters can be posted in 4 letter boxes so that all the three letters are not posted in the same letter box, is

a) 56 b) 64 c) 60 d) 72 

79) The number of different ways in which the letters of the word CONSTANT can be arranged without changing the relative positions of vowels and consonants is

a) 720 b) 1440 c) 5040 d) 360

80) If x and y are non-negative integers and x+ y=n, then which one of the following relation is true ?

a) ⁿCₓ₋₁ = ⁿₐ₋₁ b) ⁿCₓ = ⁿCₐ c) ⁿCₓ₊₁ =ⁿCₐ₊₁ d) ⁿCₓ/₂ = ⁿCₐ/₂

81) ˣCₙ = k. ˣCₙ₋₁,  then the value of k is 

a) (m - n +1)/n b) (m - n)/n  c) (m - n -1)/(n+1) d) (m - n)/(n+1)

82) In how many different ways 6 faces of a die can be coloured by 6 different colours ?

a) 6! b) 1 c) 6 d) 5 ! 

83) ther number of numbers greater than 3999 and less than 7000 which can be formed with the digit 0, 4, 5, 6, 7 (repetition of digits being allowed) is

a) 375 b) 275 c) 192 d) 328 

84) If a,b,c,d..... are (n+1) different prime numbers, then the number of different factors of aᵐbcd.... is

a) 2ᵐ⁺ⁿ -1 b) 2ᵐ⁺ⁿ c) (m+1)2ⁿ d) (m+1)2ⁿ -1

85) If n is a positive integers then which one of the following relations is true?

a) ⁿPₙ = 2. ⁿPₙ₋₂ b) ⁿPₙ = n. ⁿPₙ₋₂ c) ⁿPₙ = ⁿPₙ₋₂

86) If m parallel lines in a plane are intersected by a family of another n parallel lines, then the number of parallelograms formed in the network, is

a) mn/4 b) 1/2  mn (m -1)(n -1) c) 1/4 mn (m -1)(n -1) d)  1/4  (m -1)(n -1) 

87) The number of different ways in which three numbers can be selected from first 20 natural numbers so that the selected numbers are not consecutive, is 

a) 1124 b) 1122  c) 1120 d) 1118

88) If  n> 0 and (n²- n) C 4 = (n²- n) C 2, then value of n is

a) 2 b) 4 c) 5 d) 3

89) The value of ⁿCₓ +2 . ⁿCₓ₋₁ + ⁿCₓ₋₂ is

a) ⁿ⁺²Cₓ b) ⁿ⁺¹Cₓ c) ⁿ⁺²Cₓ₊₁ d) ⁿ⁺¹Cₓ₊₁

90) In a football tournament each team plays one match against each of the other teams and in all 153 matches are played. Then the number of teams participating in the tournament is

a) 18 b) 17 c) 19 d) 16

91) (n+1) white and (n+1) black balls are marked by numbers from 1 to (n + 1) numbers. Number of different ways in which these balls can be arranged in a row so that the two consecutive balls are of different colours, is

a) (n+1)! (n+2)! 

b) 2. (n+1)! (n+2)! 

c) ((n+1)!)² d) 2{(n+1)!}²

92)  In an examination, a student is given 3 multiple choice questions where each questions has four alternatives. Then, the number of different ways in which student cannot answer all the three questions correctly, is

a) 26  b) 27 c) 63 d) 12

93) Product of r consecutive natural numbers is always divisible r

a) r! b) (r+4)! c) (r+1)! d) (r+2)!

94) A polygon has 44 diagonals. The number of its sides is

a) 10 b) 11 c) 12 d) 13 

95) The number of permutations by taking all the letters and keeping the vowels of the word COMBINE in the odd places is

a) 96 b) 144 c) 512 d) 576

96) How many different permutations can be made with the letters of the word DROUGHT , so that the vowels are always together ?

a) 1440 b) 720 c) 2880 d) 480

97) How many different arrangement of the letters of the word BENGAL can be made so that two vowels do not come together ?

a) 720 b) 1440 c) 480 d) 360 

98) In how many ways can the letter of the word STRANGE be arranged so that the vowels may appear in the odd places ?

a) 480  b) 2880 c) 144 d) 1440

99) In how many ways can 20 first year's students and 16 second year's students be arranged so that no two second year's students may sit together 

a) ²¹O₁₆ b) (20)!. ²¹P₁₆ c) (10)!.²¹P₁₆ d) ²⁰P₁₆

100) In how many different ways can the letters of the word FOOTBALL be arranged so that two O's do not come together?

a) 7560 b) 756 c) 3780 d) 378

101) How many numbers lying between 3000 to 4000 can be formed with the digits 0, 1, 2, 3, 4 ? (repetition of the digits being allowed)

a) 120 b) 124  c) 125 d) 1024 

102) In how many different ways can 9 men be selected from 15 men so as always to exclude 3 particular men?

a) 220 b) 5002 c) 924 d) 5005

103) A man has 6 friends. In how many ways can he invite one or more of them to a party ?

a) 63 b) 64 c) 720 d) 65 

104) How many different factors can be 2310 have ?

a) 10  b) 23 c) 231  d) 31 

105) In how many ways can the crew of an 8 oared boat be arranged if two of the crew can row only on the stroke side and one can row only on the bow side .

a) 52 b) 5760 c) 112 d) 1344