Permutations and Combinations — practice questions

79 curated practice questions from Permutations and Combinations (Mathematics JEE Advanced, NTA Exams) — 31 medium, 48 hard. Every question includes the final answer free; step-by-step worked solutions with a free account.

• Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the…
• Let p,q ∈ 1,2,3, 4 . The number of equations of the form px 2 +qx + 1 = 0 having real roots must be
• The maximum number of permutations of 2n letters in which there are only a’s and b’s, taken all at a time is given by :
• The number of numbers divisible by 3 that can be formed by four different even digits is :
• Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how…
• A teacher takes 3 children from her class to the zoo at a time as often as she can, but does not take the same three…
• Let S 1 = ( i, j, k ) : i , j , k ∈ 1, 2, ...., 10 , S 2 = ( i, j ) : 1 ≤ i j + 2 ≤ 10, i , j ∈ 1, 2, ...., 10 , S 3 =…
• The sides AB, BC and CA of a triangle ABC have a, b and c interior points on them respectively, then find the number of…
• The number of ways in which five different books to be distributed among 3 persons so that each person gets at least…
• There are n different books and p copies of each in a library. The number of ways in which one or more books can be…
• The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only…
• A bag contains 2 Apples, 3 Oranges and 4 Bananas. The number of ways in which 3 fruits can be selected if atleast one…
• Match the Column The correct matching is :
• The number of ways in which the sum of upper faces of four distinct dices can be six.
• Let a n denote the number of all n -digit positive integers formed by the digits 0, 1 or both such that no consecutive…
• Three randomly chosen non-negative integers x , y and z are found to satisfy the equation x + y + z = 10 . Then the…
• An ice cream parlour has ice creams in eight different varieties. Number of ways of choosing 3 ice creams taking…
• The number of 4–digit numbers that can be made with the digits 1, 2, 3, 4 and 5 in which at least two digits are…
• 5 balls are to be placed in 3 boxes. Each box can hold all the 5 balls. Number of ways in which the balls can be placed…
• The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is
• The number of ways of distributing 10 different books among 4 students (S 1 – S 4 ) such that S 1 and S 2 get 2 books…
• If P (n, n) denotes the number of permutations of n different things taken all at a time then P (n, n) is also…
• Every one of the 10 available lamps can be switched on to illuminate certain Hall. The total number of ways in which…
• The number of subsets of the set A = a 1 , a 2 , . . . . , a n which contain even number of elements is
• Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the…
• An n -digit number is a positive number with exactly n digits. Nine hundred distinct n -digit numbers are to be formed…
• The value of 47 C 4 + _ j=1 ^ 5 ^ (52-j) C_ 3 is equal to
• If 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide the 3n…
• Number of divisors of the form 4 n + 2( n ≥ 0) of the integer 240 is
• The minimum marks required for clearing a certain screening paper is 210 out of 300. The screening paper consists of…
• Find number of arangements of 4 letters taken from the word EXAMINATION.
• The total number of six digit numbers x 1 x 2 x 3 x 4 x 5 x 6 having the property x 1 2 ≤ x 3 4 5 ≤ x 6 , is equal to:
• If the letters of the word mother are arranged in all possible orders and these words are written as in a dictionary…
• There are 20 questions in a questions paper. If no two students Explanation: e the same combination of questions but…
• If n C r denotes the number of combinations of n things taken r at a time, then the expression n C r + 1 + n C r – 1 +…
• The number of all possible selections of one or more questions from 10 given questions, each question having an…
• How many different nine digits numbers can be formed from the number 223355888 by rearranging its digits so that the…
• The number of ways in which n different prizes can be distributed amongst m (< n) persons if each is entitled to…
• The integers from 1 to 1000 are written in order around a circle. Starting at 1, every fifteenth number is marked (that…
• Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three…
• A five digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total…
• A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five…
• 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…
• The combinatorial coefficient n–1 C p denotes
• The number of times of the digits 3 will be written when listing the integer from 1 to 1000 is :
• There are five students S 1 , S 2 , S 3 , S 4 and S 5 in a music class and for them there are five seats R 1 , R 2 , R…
• One hundred identical coins, each with probability, p , of showing up heads are tossed once. If 0 p p is
• The continued product, 2 . 6 . 10 . 14 ........ to n factors is equal to :
• Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope…
• The letters of the word SURITI are written in all possible orders and these words are written out as in a dictionary…
• If n – 1 C r = ( k 2 – 3) n C r + 1 then k ∈
• The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person…
• Total 5–digit numbers divisible by 4 that can be formed using 0, 1, 2, 3, 4, 5, when the repetition of digits is…
• A variable name in a certain computer language must be either an alphabet or a alphabet followed by a decimal digit…
• The number of arrangements of two letters of the word BANANA in which two N’s do not appear adjacently is
• A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the…
• There are n concurrent lines and another line parallel to one of them. The number of different triangles that will be…
• In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number…
• The number of ways in which we can choose 3 squares of unit area on a chess board such that one of the squares has its…
• A rectangle with sides 2 m – 1 and 2 n – 1 is divided into squares of unit length by drawing parallel lines as shown in…
• Number of ways in which 3 numbers in A.P. can be selected from 1, 2, 3, ...... n is :
• Let S = 1, 2, 3, 4 . The total number of unordered pairs of disjoint subsets of S is equal to
• Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are…
• The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an…
• Two lines intersect at O. Points A 1 , A 2 . . ., A n are taken on one of them and B 1 , B 2 , . . . , B n on the…
• Let T n denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If T n…
• Two classrooms A and B having capacity of 25 and (n – 25) seats respectively. A n denotes the number of possible…
• Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of…
• There are 100 different books in a shelf. Number of ways in which 3 books can be selected so that no two of which are…
• The correct matching is :
• There are 10 questions, each question is either True or False. Number of different sequences of incorrect answers is…
• Four couples (husband and wife) decide to form a committee of four members. The number of different committee that can…
• If the L.C.M. of p , q is r 2 t 4 s 2 , where r , s , t are prime numbers and p , q are the positive integers then the…
• The value of the expression 47 C 4 + _ j =1 ^ 5 ^ 2- j C _ 3 is equal to
• The number of ways in which 10 candidates A 1 , A 2 , . . ., A 10 can be ranked so that A 1 is always before A 2 is :
• How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, repetition…
• There are 12 books on Algebra and Calculus in our library, the books of the same subject being different. If the number…
• Let S 1, 2, 3, ..., 9 . For k = 1, 2, ..., 5, let N k be the number of subsets of S , each containing five elements out…
• The number of divisors of 2 3 . 3 3 . 5 3 . 7 5 of the form 4n + 1, n ∈ N is :

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