Permutations & Combinations Questions (855)

If the ratio \({}^{2n}C_3 : {}^nC_3\) is equal to 11:1, find n.

The value of \({}^{50}C_4 + \displaystyle\sum_{r=1}^{6} {}^{56-r}C_3\) is

Let \(n\) be the number of sides of a regular polygon. If the number of diagonals of the polygon is 54, then the number of sides \(n\) is:

Find the value of r, if \(^{22}P_{r+1} : \,^{20}P_{r+2} = 11 : 52\)

\(m\) men and \(n\) women are to be seated in a row so that no two women sit together. If \(m > n\), then find the number of ways in which they can be seated.

If \(m\) parallel lines in a plane are intersected by a family of \(n\) parallel lines, the number of parallelograms that can be formed is

Find n, if (n + 2)! = 60 × (n - 1)!

Find the number of ways in which we can get a score of 11 by throwing three dice.

Given 2n different objects arranged around a circle, the number of ways of choosing k of them so that no two of them are consecutive is equal to (where 0 \leq k \leq n)

If n is even, the number of ways of arranging n persons if each person has exactly one neighbor is

In how many ways can a team of 6 horses be selected out of a stud of 16, so that there shall always be three out of \(A\ B\ C\ A'\ B'\ C'\), but never \(A\ A'\), \(B\ B'\) or \(C\ C'\) together

The set \(S = \{1, 2, 3, \ldots, 12\}\) is to be partitioned into three sets \(A, B, C\) of equal size. Thus, \(A \cup B \cup C = S\), \(A \cap B = B \cap C = A \cap C = \varnothing\). The number of ways to partition \(S\) is

The letters of the word 'CHENNAI' are arranged in all possible ways as in a dictionary, then rank of the word 'CHENNAI' from last is

Find the number of positive integral solutions satisfying the equation \((x_1 + x_2 + x_3)(y_1 + y_2) = 77\).

Number of ways of selecting one or more balls from 10 white, 9 green, and 7 black balls is:

How many 4-letter words, with or without meaning, can be formed out of the letters in the word LOGARITHMS, if repetition of letters is not allowed?

If there are n points in a plane such that no three are collinear, and p = nC₂, then find the total number of points of intersection of the lines joining these n points.

Let N be a natural number. If its first digit (from the left) is deleted, it gets reduced to \(\frac{N}{29}\). The sum of all the digits of N is

Find the number of ways in which four distinct balls can be kept into two identical boxes so that no box remains empty.

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then

Among 10 persons, \(A\), \(B\), \(C\) are to speak at a function. The number of ways in which it can be done if \(A\) wants to speak before \(B\) and \(B\) wants to speak before \(C\) is

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways Anil's mother can offer 5 fruits to Anil is

Ajay writes letters to his five friends and addresses the corresponding envelopes. The number of ways the letters can be placed in the envelopes so that at least two of them are in the wrong envelopes is

Two vowels are never together. In how many ways can we arrange them?

Find the rank of the word KRISNA when all permutations of the letters K, R, I, S, N, A are arranged in alphabetical order.

The total number of ways in which \(n^2\) number of identical balls can be put in \(n\) numbered boxes \((1, 2, 3, \ldots, n)\) such that \(i^{\text{th}}\) box contains at least \(i\) number of balls is

Find the number of permutations of all the letters of the word MATHEMATICS which starts with consonants only.

The total number of 3-digit numbers whose sum of digits is 10 is ________. (JEE Main 2020)

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is

Let n be a four-digit integer in which all the digits are different. If x is number of odd integers and y is number of even integers, then

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

\(\frac{100}{10!}=\frac{1}{8!}+\frac{x}{9!}\). Find x .

Find the number of words which can be formed using all the letters of the word 'INSTITUTION' which start with consonant.

How many words can be made with the letters of the word INSTITUTION such that vowels and consonants alternate?

How many different signals can be given using any number of flags from 5 flags of different colors?

In how many ways can ₹16 be divided into 4 persons when none of them gets less than ₹3?

The number of four letter words that can be formed using the letters of the word BARRACK is

In how many ways can first and second prizes in Mathematics, first and second prizes in Physics, first prize in Chemistry and first prize in English be given away to a class of 30 students?

Two different packs of cards are shuffled together. Cards are dealt equally among 4 players, each getting 13 cards. In how many ways can a player get his cards if no two cards are from the same suit with the same denomination (i.e., two cards are identical), 13 cards are to be selected from 52 cards where each card is two in number?

The sum of the odd divisors of $10!$ is

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is

A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least two balls of each color?

The number of words of four letters containing equal number of vowels and consonants, where repetition is allowed, is

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is

The total number of ways of selecting two numbers from the set \(\{1, 2, 3, 4, \ldots, 3n\}\) so that their sum is divisible by 3 is equal to

In how many ways can 3 boys and 15 girls sit together in a row such that between any 2 boys at least 2 girls sit?

There are n married couples at a party. Each person shakes hand with every person other than her or his spouse. Find the total number of hand-shakes.

Find the number of positive integral solutions of \(xyz = 21600\).

How many line segments have both their endpoints located at the vertices of a given cube.

The number of ways, in which 5 boys and 3 girls can be seated on a round table if a particular boy \(B_1\) and a particular girl \(G_1\) never sit adjacent to each other, is