Permutations & Combinations Questions (855)

If \(^n C_4, \ ^nC_5 \ and \ ^n C_6\) are in AP, then n can be

Find the number of non-negative integral solutions of \(x + y + z + w \leq 20\).

From an unlimited number of red, white, blue and green balls, a selection of 18 balls is to be made so that there are at least two of each colour. If the number of selection is \(k\), then \(k\) is equal to:

Let A = {1, 2, 3, 4, 5, 6, 7}. Find the total number of functions f : A → A such that f(f(x)) = x for all x ∈ A (i.e., involutions on A). The answer is 351.

The number of numbers formed by nine non-zero 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

Let \(n_1

How many selections of atleast one red ball can be made from a bag containing 4 red balls and 5 black balls, where balls of the same colour are identical?

Number of ways in which 12 different books can be distributed equally among 3 persons, is

There are 5 historical monuments, 6 gardens, and 7 shopping malls in a city. In how many ways a tourist can visit the city if he visits at least one shopping mall?

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

Sum of all 4-digit numbers using digits 2,1,2,3 (all used) is equal to ____.

Number of permutations of 1,2,3,…,7 without repetition, containing neither string 153 nor 2467, is _______.

Number of 4-letter words (2 vowels, 2 consonants) from UNIVERSE without repetition is _____.

The total number of selections of at most 4 things from 9 different things is ______.

If the best and the worst papers never appear together, find in how many ways six examination papers can be arranged.

A hall has 12 gates. In how many ways can a man enter the hall through one gate and come out through a different gate?

8 persons transported in 3 cars (at most 3 each). Number of ways is

The total number of positive integral solutions of \(15

Total 3-digit numbers divisible by 6, using $\{1,2,3,4,5\}$ with repetition, is ________

Let f : A → A be an invertible function where A = {1, 2, 3, 4, 5, 6}. Then find the number of these functions in which at least three elements have self-image.

Given that P(n): 2n < n!, ∀n ∈ N. For which value of n is P(n) first true? The value of the filler is:

The number of distinct natural numbers up to a maximum of four digits and divisible by 5, which can be formed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, each digit not occurring more than once in each number, is

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is _____

Number of quadrilaterals that can be formed using the vertices of a polygon of sides 'n' if exactly 1 side of the quadrilateral is common with side of the n-gon, is

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

A box contains two white balls, three black balls and four red balls. The number of ways in which three balls can be drawn from the box, so that at least one of the balls is black, is

The number of ways of arranging 5 boys and 3 girls on a round table such that boys \(B_1\) and girl \(G_1\) are never together is:

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is

The number of words which can be formed using all the letters of the word ``DAUGHTER'' so that all the vowels never come together, is:

Number of arrangements of INDEPENDENCE with all vowels always together is

In a mixed doubles tournament, if matches where no couple plays together $=840$, then total participants is ________.

Digits $a,b,c$ in AP. 9-digit numbers using each thrice with at least one set of 3 consecutive digits in AP. How many such numbers?

A 4-digit PIN: all digits different, greatest digit $=7$, sum of first two $=$ sum of last two. Maximum trials needed is ________.

Number of triplets $(x,y,z)$ with distinct non-negative integers and $x+y+z=15$ is

A bag contains four one-rupee coins, two twenty-five paisa coins, and five ten-paisa coins. In how many ways can an amount, not less than ₹1 be taken out from the bag? (Consider coins of the same denominations to be identical.)

The number of ways in which we can arrange the 2n students with n boys \(b_1, b_2, \ldots, b_n\) and n girls \(g_1, g_2, \ldots, g_n\) in a line so that all the boys and all the girls stand in increasing order of their age (Assume they all are of different age)

Mr. Anshuman has thrown a dice 6 times in how many ways we can get a sum greater than 17?

In an election, the number of candidates exceeds the number to be elected by 2. A man can vote in 56 ways. Find the number of candidates.

All letters of PUBLIC written in all possible orders (dictionary). Serial number of PUBLIC is

5 students seated in 5 allotted seats. Number of ways none sits on allotted seat is

If a number of arrangements of the letters of the word MONOTONICITY in which O’s do not appear adjacently is 9! (k), then k equals:

A staircase has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1's and 2's. If the number of ways in which the person can go up the stairs is $p$, then find $\frac{p}{89}$

There are $n$ persons sitting around a circular table. They start singing a 2 minute song in pairs such that no two persons sitting together will sing together. This process is continued for 28 minutes. Find $n$

The number of 5-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$ (repetition allowed) such that the sum of the first and last digit is at most 8 is

If the number of ways in which 8 people can be arranged in a line if $A$ and $B$ must be next to each other and $C$ must be somewhere behind $D$ is equal to $'m'$ then sum of all the digits of $m$ is equal to ______.

There are three stations A, B and C, five routes for going from station A to station B and four routes for going from station B to station C. Find the number of different ways through which a person can go from A to C via B.

Six $X$'s have to be placed in the squares of the figure given below, such that each row contains at least one X. If the total no. of different ways this can be done is $'m'$ then $\frac{m}{13}$ is equal to ________.

A conference attended by 200 delegates is held in a hall. The hall has 7 doors, marked $A, B, \ldots \ldots, G$. At each door, an entry book is kept and the delegates entering through that door sign it in the order in which they enter. If each delegate is free to enter any time through and through any door he likes, if the total no. of different sets of seven lists would arise in all is equal to $''P_r''$ then $'n-r'$ is equal to (Assume that every person signs only at his first entry).

The number of ways to choose 5 letters from the English alphabet in increasing alphabetical order such that the middle (3rd) letter is M is

In how many rotationally distinct ways can the vertices of a cube be coloured with black or white colour?