Permutations and Combinations — JEE Main PYQs

The number of strictly increasing functions f from the set $\{1,2,3,4,5,6\}$ to the set $(1,2,3,\dots ,9)$ such that $\mathrm{f}(\mathrm{i})\ne \mathrm{i}$ for $1\le \mathrm{i}\le 6$, is equal to :

Let $\mathrm{S}=\{(\mathrm{m},\mathrm{n}):\mathrm{m},\mathrm{n}\in \{1,2,3,\dots ..,50\}\}$. If the number of elements ( $\mathrm{m},\mathrm{n}$ ) in S such that $6^{\mathrm{m}}+9^{\mathrm{n}}$ is a multiple of 5 is p and the number of elements …

The largest $\mathrm{n}\in \mathrm{N}$, for which $7^{\mathrm{n}}$ divides 101!, is:

Let ABC be a triangle. Consider four points ${\mathrm{p}}_1,{\mathrm{p}}_2$, ${\mathrm{p}}_3,{\mathrm{p}}_4$ on the side AB , five points ${\mathrm{p}}_5,{\mathrm{p}}_6,{\mathrm{p}}_7,{\mathrm{p}}_8,{\mathrm{p}}_9$ on the side BC and four points …

Let S be the set of the first 11 natural numbers. Then the number of elements in $A=\{B\subseteq S:n(B)\ge 2$ and the product of all elements of B is even $\}$ is

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ______.

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

Let S denote the set of 4-digit numbers abcd such that $a>b>c>d$ and P denote the set of 5-digit numbers having product of its digits equal to 20. Then $n(S)+n(P)$ is equal to...........

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits $0,1,2,5,9$, if the repetition of the digits is allowed, is ______

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is :

The largest value of n, for which $40^{\mathrm{n}}$ divides 60!, is

Let $\mathrm{S}=\{1,2,3,4,5,6,7,8,9\}$. Let x be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only …

Let $S=\left\{x^3+a{x}^2+bx+c:a,b,c\in N\text{ and }a,b,c\le 20\right\}$ be a set of polynomials. Then the number of polynomials in S , which are divisible by ${\mathrm{x}}^2+2$, is

Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text{th }}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ______。

The number of seven-digit numbers, that can be formed by using the digits $1,2,3,5$ and $7$ such that each digit is used at least once, is :

Let $\mathrm{A}=\{(\mathrm{a},\mathrm{b},\mathrm{c}):\mathrm{a},\mathrm{b},\mathrm{c}$ are non-negative integers and $\mathrm{a}+\mathrm{b}+2\mathrm{c}=22\}$.
The $\mathrm{n}(\mathrm{A})$ is equal to :

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

Two players A and B play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player A wins the series is ______.

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is:

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text{th }}$ floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does …

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is

Let ${\mathrm{p}}_{\mathrm{n}}$ denote the total number of triangles formed by joining the vertices of an n-side regular polygon. If $p_{n+1}-p_n=66$, then the sum of all distinct prime divisors of n is :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ are not adjacent to each …

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

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is ______.

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{\text{th }}$ position in this arrangement, is :

Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set P is :

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S , one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15 , is

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0,1, $2,3,4,5,6,7$, such that the sum of their first and last digits should not be more than 8 , is

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B , is equal to:

Number of functions $f:\{1,2,\dots ,100\}\to \{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ____.

The number of 3-digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is

Let $\mathrm{S}=\left\{{\mathrm{p}}_1,{\mathrm{p}}_2\dots \dots ,{\mathrm{p}}_{10}\right\}$ be the set of first ten prime numbers. Let $\mathrm{A}=\mathrm{S}\cup \mathrm{P}$, where P is the set of all possible products of distinct element of S . Then the number of all ordered …

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin O . In the first quadrant, ${\mathrm{P}}_1,{\mathrm{P}}_2,\dots .{\mathrm{P}}_{12}$ are 12 points on line ${\mathrm{L}}_1$ and …

The largest $n\in \mathbb{N}$ such that $3^n$ divides 50! is:

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by ${\mathrm{W}}_{\mathrm{n}}$. Let the probability $\mathrm{P}\left({\mathrm{W}}_{\mathrm{n}}\right)$ o …

If the number of seven-digit numbers, such that the sum of their digits is even, is $m\cdot n\cdot 10^{\mathrm{n}}$; $m,n\in \{1,2,3,\dots ,9\}$, then $m+n$ is equal to

Let m and $\mathrm{n},(\mathrm{m}<\mathrm{n})$ be two 2-digit numbers. Then the total numbers of pairs $(m,n)$, such that $\gcd (m,n)=6$, is

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P . If $\mathrm{p}+\mathrm{q}=126$, then the eccentricity of the ellipse …

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways su …

For $n\ge 2$, let $S_n$ denote the set of all subsets of $\{1,2,\dots ,n\}$ with no two consecutive numbers. For example $\{1,3,5\}\in S_6$, but $\{1,2,4\}\notin S_6$. Then $n(S_5)$ is equal to

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

There are 5 points $P_1,P_2,P_3,P_4,P_5$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $P_6,P_7,\dots ,P_{11}$ on the side BC and 7 points $P_{12},P_{13},\dots ,P_{18}$ on the side CA of the triangle. The number o …

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ____.