JEE Mathematics Permutation and Combination MCQs Set 06

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

Question. A cabinet of ministers consists of 11 ministers, one minister being the chief minister. A meeting is to be held in a room having a round table and 11 chairs round it, one of them being meant for the chairman. The number of ways in which the ministers can take their chairs, the chief minister occupying the chairman’s place, is
(a) \( \frac{1}{2}(10!) \)
(b) \( 9! \)
(c) \( 10! \)
(d) None of the options
Answer: (c) \( 10! \)

Question. The number of ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats is
(a) 1440
(b) 720
(c) 5040
(d) None of the options
Answer: (a) 1440

Question. The number of ways in which 20 different pearls of two colours can be set alternately on a necklace, there being 10 pearls of each colour, is
(a) \( 9! \times 10! \)
(b) \( 5(9!)^2 \)
(c) \( (9!)^2 \)
(d) None of the options
Answer: (b) \( 5(9!)^2 \)

Question. If \( r > p > q \), the number of different selections of \( p + q \) things taking \( r \) at a time where \( p \) things are identical and \( q \) other things are identical, is
(a) \( p + q - r \)
(b) \( p + q - r + 1 \)
(c) \( r - p - q + 1 \)
(d) None of the options
Answer: (b) \( p + q - r + 1 \)

Question. There are 4 mangoes, 3 apples, 2 oranges and 1 each of 3 other verieties of fruits. The number of ways of selecting at least one fruit of each king is
(a) 10!
(b) 9!
(c) 4!
(d) None of the options
Answer: (c) 4!

Question. The number of proper divisors of \( 2^p \cdot 6^q \cdot 15^r \) is
(a) \( (p + q + 1)(q + r + 1)(r + 1) \)
(b) \( (p + q + 1)(q + r + 1)(r + 1) - 2 \)
(c) \( (p + q)(q + r)r - 2 \)
(d) None of the options
Answer: (b) \( (p + q + 1)(q + r + 1)(r + 1) - 2 \)

Question. The number of proper divisors of 1800 which are also divisible by 10, is
(a) 18
(b) 34
(c) 27
(d) None of the options
Answer: (a) 18

Question. The number of odd proper divisors of \( 3^p \cdot 6^m \cdot 21^n \) is
(a) \( (p + 1)(m + 1)(n + 1) - 2 \)
(b) \( (p + m + n + 1)(n + 1) - 1 \)
(c) \( (p + 1)(m + 1)(n + 1) - 1 \)
(d) None of the options
Answer: (b) \( (p + m + n + 1)(n + 1) - 1 \)

Question. The number of even proper divisors of 1008 is
(a) 23
(b) 24
(c) 22
(d) None of the options
Answer: (a) 23

Question. In a test there were n questions. In the test \( 2^{n-i} \) students gave wrong answers to i questions where \( i = 1, 2, 3, \dots, n \). If the total number of wrong answers given is 2047 then n is
(a) 12
(b) 11
(c) 10
(d) None of the options
Answer: (b) 11

Question. The number of ways to give 16 different things to three persons A, B, C so that B gets 1 more than A and C gets 2 more than B, is
(a) \( \frac{16!}{4!5!7!} \)
(b) \( 4! \cdot 5! \cdot 7! \)
(c) \( \frac{16!}{3!5!8!} \)
(d) None of the options
Answer: (a) \( \frac{16!}{4!5!7!} \)

Question. The number of ways to distribute 32 different things equally among 4 persons is
(a) \( \frac{32!}{(8!)^3} \)
(b) \( \frac{32!}{(8!)^4} \)
(c) \( \frac{1}{4}(32!) \)
(d) None of the options
Answer: (b) \( \frac{32!}{(8!)^4} \)

Question. If 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide the 3n things in 3 equal groups is
(a) \( k \times 3! \)
(b) \( \frac{k}{3!} \)
(c) \( (3!)^k \)
(d) None of the options
Answer: (b) \( \frac{k}{3!} \)

Question. In a packet there are m different books, n different pens and p different pencils. The number of selections of at least one article of each type from the packet is
(a) \( 2^{m+n+p} - 1 \)
(b) \( (m + 1)(n + 1)(p + 1) - 1 \)
(c) \( 2^{m+n+p} \)
(d) None of the options
Answer: (a) \( 2^{m+n+p} - 1 \)

Question. The number of 6-digit numbers that can be made with the digits 1, 2, 3 and 4 and having exactly two pairs of digits is
(a) 480
(b) 540
(c) 1080
(d) None of the options
Answer: (c) 1080

Question. The number of words of four letters containing equal number of vowels and consonants, repetition being allowed, is
(a) \( 105^2 \)
(b) \( 210 \times 243 \)
(c) \( 105 \times 243 \)
(d) None of the options
Answer: (b) \( 210 \times 243 \)

Question. The number of ways in which 6 different balls can be put in two boxes of different sizes so that no box remains empty is
(a) 62
(b) 64
(c) 36
(d) None of the options
Answer: (a) 62

Question. A shopkeeper selling 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 a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the show case is
(a) 6
(b) 50
(c) 150
(d) None of the options
Answer: (c) 150

Question. The number of arrangements of the letters of the word BHARAT taking 3 at a time is
(a) 72
(b) 120
(c) 14
(d) None of the options
Answer: (a) 72

Question. The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the number is 10 and the sums of the numbers placed diagonally are equal, is
(a) \( 2! \times 2! \)
(b) \( 4! \)
(c) \( 2(4!) \)
(d) None of the options
Answer: (d) None of the options

Question. The number of positive integral solutions of \( x + y + z = n, n \in N, n \geq 3 \), is
(a) \( {}^{n-1}C_2 \)
(b) \( {}^{n-1}P_2 \)
(c) \( n(n - 1) \)
(d) None of the options
Answer: (a) \( {}^{n-1}C_2 \)

Question. The number of non-negative integral solutions of \( a + b + c + d = n, n \in N \), is
(a) \( {}^{n+3}P_2 \)
(b) \( \frac{(n + 1)(n + 2)(n + 3)}{6} \)
(c) \( {}^{n-1}C_{n-4} \)
(d) None of the options
Answer: (b) \( \frac{(n + 1)(n + 2)(n + 3)}{6} \)

Question. The number of points (x, y, z) in space, whose each coordinate is a negative integer such that \( x + y + z + 12 = 0 \), is
(a) 385
(b) 55
(c) 110
(d) None of the options
Answer: (b) 55

Question. If a, b, c are three natural number in AP and \( a + b + c = 21 \) then the possible number of values of the ordered triplet (a, b, c) is
(a) 15
(b) 14
(c) 13
(d) None of the options
Answer: (c) 13

Question. If a, b, c, d are odd natural number such that \( a + b + c + d = 20 \) then the number of values of the ordered quadruplet (a, b, c, d) is
(a) 165
(b) 455
(c) 310
(d) None of the options
Answer: (a) 165

Question. If \( x, y, z \) are integers and \( x \geq 0, y \geq 1, z \geq 2, x + y + z = 15 \) then the number of values of the ordered triplet (x, y, z) is
(a) 91
(b) 455
(c) \( {}^{17}C_{15} \)
(d) None of the options
Answer: (a) 91

Question. If a, b, c are positive integers such that \( a + b + c \leq 8 \) then the number of possible values of the ordered triplet (a, b, c) is
(a) 84
(b) 56
(c) 83
(d) None of the options
Answer: (b) 56

Question. The number of different ways of distributing 10 marks among 3 questions, each question carrying at least 1 mark, is
(a) 72
(b) 71
(c) 36
(d) None of the options
Answer: (c) 36

Question. The number of ways to give away 20 apples to 3 boys, each boy receiving at least 4 apples, is
(a) \( {}^{10}C_8 \)
(b) 90
(c) \( {}^{22}C_{20} \)
(d) None of the options
Answer: (a) \( {}^{10}C_8 \)

Question. The position vector of a point P is \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \), where \( x \in N, y \in N, z \in N \) and \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \). If \( \vec{r} \cdot \vec{a} = 10 \), the number of possible positions of P is
(a) 36
(b) 72
(c) 66
(d) None of the options
Answer: (a) 36

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

Question. There are n seats round a table numbered 1, 2, 3, \dots, n. The number of ways in which \( m (\leq n) \) persons can take seats is
(a) \( {}^nP_m \)
(b) \( {}^nC_m \times (m - 1)! \)
(c) \( {}^{n-1}P_{m-1} \)
(d) \( {}^nC_m \times m! \)
Answer: (a) \( {}^{nP}_m \) and (d) \( {}^nC_m \times m! \)

Question. Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and let \( \vec{r} \) be a variable vector such that \( \vec{r} \cdot \hat{i}, \vec{r} \cdot \hat{j} \) and \( \vec{r} \cdot \hat{k} \) are positive integers. If \( \vec{r} \cdot \vec{a} \leq 12 \) then the number of values of \( \vec{r} \) is
(a) \( {}^{12}C_9 - 1 \)
(b) \( {}^{12}C_3 \)
(c) \( {}^{12}C_9 \)
(d) None of the options
Answer: (b) \( {}^{12}C_3 \) and (c) \( {}^{12}C_9 \)

Question. The total number of ways in which a beggar can be given at least one rupee from four 25-paisa coins, three 50-paisa coins and 2 one-rupee coins, is
(a) 54
(b) 53
(c) 51
(d) None of the options
Answer: (a) 54

Question. For the equation \( x + y + z + w = 19 \), the number of positive integral solutions is equal to
(a) the number of ways in which 15 identical things can be distributed among 4 persons
(b) the number of ways in which 19 identical things can be distributed among 4 persons
(c) coefficient of \( x^{19} \) in \( (x^0 + x^1 + x^2 + \dots + x^{19})^4 \)
(d) coefficient of \( x^{19} \) in \( (x + x^2 + x^3 + \dots + x^{19})^4 \)
Answer: (a) the number of ways in which 15 identical things can be distributed among 4 persons and (d) coefficient of \( x^{19} \) in \( (x + x^2 + x^3 + \dots + x^{19})^4 \)