Permutations and Combinations — JEE Main PYQs

Let $\mathrm{f}(\mathrm{x})=2{\mathrm{x}}^{\mathrm{n}}+\lambda ,\lambda \in \mathbb{R},\mathrm{n}\in \mathbb{N}$, and $\mathrm{f}(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of ( $\mathrm{f}(3)-\mathrm{f}(2)$ ) is

The number of numbers, strictly between 5000 and 10000 can be formed using the digits $1,3,5,7,9$ without repetition, 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 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can of …

The number of integers, greater than 7000 that can be formed, using the digits $3,5,6,7,8$ without repetition, is

The number of square matrices of order 5 with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

Let $\mathrm{A}=[{\mathrm{a}}_{\mathrm{ij}}],{\mathrm{a}}_{\mathrm{ij}}\in \mathrm{Z}\cap [0,4],1\le \mathrm{i},\mathrm{j}\le 2$. The number of matrices A such that the sum of all entries is a prime number $\mathrm{p}\in (2,13)$ is ______.

The number of ways of selecting two numbers $a$ and $b$, $a\in \{2,4,6,\dots ,100\}$ and $b\in \{1,3,5,\dots ,99\}$ such that 2 is the remainder when $\mathrm{a}+\mathrm{b}$ is divided by 23 is

The number of seven digits odd numbers, that can be formed using all the seven digits 1,2,2,2,3,3,5 is ____.

Let 5 digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is ______.

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ____.

Let $\mathrm{S}=\{1,2,3,5,7,10,11\}$. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is ______.

Let x and y be distinct integers where $1\le \mathrm{x}\le 25$ and $1\le y\le 25$. Then, the number of ways of choosing x and y, such that $\mathrm{x}+\mathrm{y}$ is divisible by 5, is ______.

If ${}^{2n+1}{P}_{n-1}:{}^{2n-1}{P}_n=11:21$, then $n^2+n+15$ is equal to ______

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by 55 is:

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D},\mathrm{E}\}$ or a number from $\{1,2,3,4,5\}$ with the repetition of characters allowed. If the number of …

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $\mathrm{b}+3\mathrm{g}$ is equal to

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ____.

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits $1,2,3,4,5$ and 6 without repetition of digits. Then the total number of such numbers is

The number of bijective functions $\mathrm{f}:\{1,3,5,7,\dots ,99\}\to \{2,4,6,8,\dots ,100\}$, such that $\mathrm{f}(3)\ge \mathrm{f}(9)\ge \mathrm{f}(15)\ge \mathrm{f}(21)\ge \dots \ge \mathrm{f}(99)$, is

The number of 5-digit natural numbers, such that the product of their digits is 36, is

The number of ways to distribute 30 identical candies among four children ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3$ and ${\mathrm{C}}_4$ so that ${\mathrm{C}}_2$ receives atleast 4 and atmost 7 candies, ${\mathrm{C}}_3$ receives atleast 2 and atmost 6 candies, is e …

Let $\mathrm{S}=\{1,2,3,4\}$. Then the number of elements in the set $\{f:S\times S\to S:f$ is onto and $f(a,b)=f(b,a)\ge \mathrm{a}\forall (\mathrm{a},\mathrm{b})\in \mathrm{S}\times \mathrm{S}\}$ is

Let A be a matrix of order $2\times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p$, $2<p<8$, then the number of such matrices A is :

The total number of 5-digit numbers, formed by using the digits $1,2,3,5,6,7$ without repetition, which are multiple of 6, is

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ____.

Let ${\mathrm{b}}_1{\mathrm{b}}_2{\mathrm{b}}_3{\mathrm{b}}_4$ be a 4-element permutation with ${\mathrm{b}}_{\mathrm{i}}\in \{1,2,3,\dots \dots \dots ,100\}$ for $1\le i\le 4$ and $b_i\ne b_j$ for $i\ne j$, such that either …

There are ten boys $B_1,B_2,\dots ,B_{10}$ and five girls $G_1,G_2,\dots ,G_5$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_1$ and $B_2$ together should not be the members of a group, is __ …

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ____.

Let $\mathrm{S}=\{4,6,9\}$ and $\mathrm{T}=\{9,10,11,\dots ,1000\}$. If …

The number of one-one function $\mathrm{f}:\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}\to \{0,1,2,\dots ,10\}$ such that $2\mathrm{f}(\mathrm{a})-\mathrm{f}(\mathrm{b})+3\mathrm{f}(\mathrm{c})+\mathrm{f}(\mathrm{d})=0$ is

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct There are 3 marks for each correct answer, -2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the …

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits $1,2,3,4,5,7$ and $9$ is ______.

The sum of all the elements of the set $\{\alpha \in \{1,2,\dots ,100\}:\mathrm{HCF}(\alpha ,24)=1\}$ is ______.

The total number of three-digit numbers, with one digit repeated exactly two times, is

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is ______.

Let A be a $3\times 3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices A having sum of all the entries equal to 5, is ____

If ${}^1{P}_1+2\cdot {}^2{P}_2+3\cdot {}^3{P}_3+\dots \dots \dots +15\cdot {}^{15}{P}_{15}={}^q{P}_r-s,0\le s\le 1$, then ${}^{q+s}{C}_{r-s}$ is equal to ______.

The number of three digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is ____.

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways the committee can be formed is:

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is ____.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ______.

The students ${\mathrm{S}}_1,{\mathrm{S}}_2,\dots \dots .{\mathrm{S}}_{10}$ are to be divided into 3 groups $\mathrm{A},\mathrm{B}$ and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of formi …

Let $S=[1,2,3,4,5,6,9]$. Then the number of elements in the set $T=\{A\subseteq S:A\ne \varphi$ and the sum of all the elements of $A$ is not a multiple of 3$\}$ is ______.

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1,2, 3,4,5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is

If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8$, then the number of all numbers greater than 10,000 is equal to ____

The sum of all the elements in the set $\{n\in \{1,2,\dots ,100\}\mid$ H.C.F. of $n$ and 2040 is 1$\}$ is equal to ____