JEE Mathematics Permutation and Combination MCQs Set 04

Choose the most appropriate option (a, b, c or d).

Question. If \( {}^nC_{r-1} = 56 \), \( {}^nC_r = 28 \) and \( {}^nC_{r+1} = 8 \) then \( r \) is equal to
(a) 8
(b) 6
(c) 5
(d) None of the options
Answer: (b) 6

Question. The value of \( {}^{20}C_{31} + \sum_{j=0}^{10} {}^{40+j}C_{10+j} \) is equal to
(a) \( {}^{51}C_{20} \)
(b) \( 2 \cdot {}^{50}C_{20} \)
(c) \( 2 \cdot {}^{45}C_{15} \)
(d) None of the options
Answer: (a) \( {}^{51}C_{20} \)

Question. In a group of boys, the number of arrangements of 4 boys is 12 times the number of arrangements of 2 boys. The number of boys in the group is
(a) 10
(b) 8
(c) 6
(d) None of the options
Answer: (c) 6

Question. The value of \( \sum_{r=1}^{10} r \cdot {}^rP_r \) is
(a) \( {}^{11}P_{11} \)
(b) \( {}^{11}P_{11} - 1 \)
(c) \( {}^{11}P_{11} + 1 \)
(d) None of the options
Answer: (b) \( {}^{11}P_{11} - 1 \)

Question. From a group of persons the number of ways of selecting 5 persons is equal to that of 8 persons. The number of persons in the group is
(a) 13
(b) 40
(c) 18
(d) 21
Answer: (a) 13

Question. The number of distinct rational numbers \( x \) such that \( 0 < x < 1 \) and \( x = \frac{p}{q} \), where \( p, q \in \{1, 2, 3, 4, 5, 6\} \), is
(a) 15
(b) 13
(c) 12
(d) 11
Answer: (d) 11

Question. The total number of 9-digit numbers of different digits is
(a) \( 10(9!) \)
(b) \( 8(9!) \)
(c) \( 9(9!) \)
(d) None of the options
Answer: (c) \( 9(9!) \)

Question. The number of 6-digit numbers that can be made with the digits 0, 1, 2, 3, 4 and 5 so that even digits occupy odd places, is
(a) 24
(b) 36
(c) 48
(d) None of the options
Answer: (a) 24

Question. The number of ways in which 6 men can be arranged in a row so that three particular men are consecutive, is
(a) \( {}^4P_4 \)
(b) \( {}^4P_4 \times {}^3P_3 \)
(c) \( {}^3P_3 \times {}^3P_3 \)
(d) None of the options
Answer: (b) \( {}^4P_4 \times {}^3P_3 \)

Question. Seven different lectures are to deliver lectures in seven periods of a class on a particular day. A, B and C are three of the lectures. The number of ways in which a routine for the day can be made such that A delivers his lecture before B, and B before C, is
(a) 420
(b) 120
(c) 210
(d) None of the options
Answer: (d) None of the options

Question. The total number of 5-digit numbers of different digits in which the digit in the middle is the largest is
(a) \( \sum_{n=4}^{9} {}^nP_4 \)
(b) \( 33(3!) \)
(c) \( 30(3!) \)
(d) None of the options
Answer: (d) None of the options

Question. A 5-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) 216
(b) 600
(c) 240
(d) 3125
Answer: (a) 216

Question. Let \( A = \{x \mid x \text{ is a prime number and } x < 30\} \). The number of different rational numbers whose numerator and denominator belong to \( A \) is
(a) 90
(b) 180
(c) 91
(d) None of the options
Answer: (c) 91

Question. The total number of ways in which six ‘+’ and four ‘-’ signs can be arranged in a line such that no two ‘-’ signs occur together is
(a) \( \frac{7!}{3!} \)
(b) \( 6! \times \frac{7!}{3!} \)
(c) 35
(d) None of the options
Answer: (c) 35

Question. The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between two consonants is
(a) 1440
(b) 1800
(c) 2160
(d) None of the options
Answer: (b) 1800

Question. The number of numbers of four different digits that can be formed from the digits of the number 12356 such that the numbers are divisible by 4, is
(a) 36
(b) 48
(c) 12
(d) 24
Answer: (a) 36

Question. Let \( S \) be the set of all functions from the set \( A \) to the set \( A \). If \( n(A) = k \) then \( n(S) \) is
(a) \( k! \)
(b) \( k^k \)
(c) \( 2^k - 1 \)
(d) \( 2^k \)
Answer: (b) \( k^k \)

Question. Let \( A \) be the set of 4-digit numbers \( a_1 a_2 a_3 a_4 \) where \( a_1 > a_2 > a_3 > a_4 \) then \( n(A) \) is equal to
(a) 126
(b) 84
(c) 210
(d) None of the options
Answer: (c) 210

Question. The number of numbers divisible by 3 that can be formed by four different even digits is
(a) 18
(b) 36
(c) 0
(d) None of the options
Answer: (b) 36

Question. The number of 5-digit even number that can be made with the digit 0, 1, 2 and 3 is
(a) 384
(b) 192
(c) 768
(d) None of the options
Answer: (a) 384

Question. The number of 4-digit numbers that can be made with the digit 1, 2, 3, 4 and 5 in which at least two digits are identical, is
(a) \( 4^5 - 5! \)
(b) 505
(c) 600
(d) None of the options
Answer: (b) 505

Question. The number of words that can be made by rearranging the letters of the word APURBA so that vowels and consonants alternate is
(a) 18
(b) 35
(c) 36
(d) None of the options
Answer: (c) 36

Question. The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with a constant, is
(a) \( \frac{5(7!)}{2} \)
(b) \( \frac{3(7!)}{2} \)
(c) \( 2(7!) \)
(d) None of the options
Answer: (a) \( \frac{5(7!)}{2} \)

Question. The number of numbers of 9 different nonzero digits such that all the digits in the first four places are less than the digit in the middle and all the digits in the last four places are greater than that in the middle is
(a) \( 2(4!) \)
(b) \( (4!)^2 \)
(c) \( 8! \)
(d) None of the options
Answer: (b) \( (4!)^2 \)

Question. In the decimal system of numeration the number of 6-digit numbers in which the digit in any place is greater than the digit to the left of it is
(a) 210
(b) 84
(c) 126
(d) None of the options
Answer: (b) 84

Question. The number of 5-digit numbers in which no two consecutive digits are identical is
(a) \( 9^2 \times 8^3 \)
(b)
(c) \( 9 \times 9^4 \)
(d) None of the options
Answer: (c) \( 9 \times 9^4 \)

Question. In the decimal system of numeration the number of 6-digit numbers in which the sum of the digits is divisible by 5 is
(a) 180000
(b) 540000
(c) \( 5 \times 10^5 \)
(d) None of the options
Answer: (a) 180000

Question. The sum of all the numbers of four different digits that can be made by using the digits 0, 1, 2 and 3 is
(a) 26664
(b) 39996
(c) 38664
(d) None of the options
Answer: (c) 38664

Question. A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is
(a) 12
(b) 10
(c) 60
(d) None of the options
Answer: (b) 10

Question. ABCD is a convex quadrilateral. 3, 4, 5 and 6 points are marked on the sides AB, BC, CD and DA respectively. The number of triangles with vertices on different sides is
(a) 270
(b) 220
(c) 282
(d) None of the options
Answer: (d) None of the options

Choose the correct options. One or more options may be coorect.

Question. If \( P = n(n^2 - 1^2)(n^2 - 2^2)(n^2 - 3^2) \dots (n^2 - r^2) \), \( n > r \), \( n \in N \), then P is divisible by
(a) \( (2r + 2)! \)
(b) \( (2r - 1)! \)
(c) \( (2r + 1)! \)
(d) None of the options
Answer: (b) \( (2r - 1)! \) and (c) \( (2r + 1)! \)

Question. If \( {}^{n+5}P_{n+1} = \frac{11(n-1)}{2} \cdot {}^{n+3}P_n \) then value of n is
(a) 7
(b) 8
(c) 6
(d) 5
Answer: (a) 7 and (c) 6

Question. If \( {}^nC_4, {}^nC_5 \) and \( {}^nC_6 \) are in AP then n is
(a) 8
(b) 9
(c) 14
(d) 7
Answer: (c) 14 and (d) 7

Question. The product of r consecutive integers is divisible by
(a) \( r \)
(b) \( \sum_{k=1}^{r-1} k \)
(c) \( r! \)
(d) None of the options
Answer: (a) \( r \), (b) \( \sum_{k=1}^{r-1} k \), and (c) \( r! \)