Permutations and Combinations — JEE Main PYQs

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 is:

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55 , is

A natural number has prime factorization given by $n=2^x{3}^y{5}^z$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6},y>z$. Then the number of odd divisors of $n$, including 1 , is :

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is ______

Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $\mathrm{g}(3)=2\mathrm{g}(1)$ is :

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ______.

Team 'A' consists of $7$ boys and $n$ girls and Team 'B' has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:

There are 5 students in class 10,6 students in class 11 and 8 students in class 12 . If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is …

If ${}^n{P}_r={}^n{P}_{r+1}$ and ${}^n{C}_r={}^n{C}_{r-1}$, then the value of $r$ is equal to:

Let n be a non-negative integer. Then the number of divisors of the form "$4\mathrm{n}+1$" of the number $(10{)}^{10}\cdot (11{)}^{10}\cdot (13{)}^{13}$ is equal to ____

Let ${\mathrm{P}}_1,{\mathrm{P}}_2\dots ,{\mathrm{P}}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points ${\mathrm{P}}_{\mathrm{i}},{\mathrm{P}}_{\mathrm{j}}$, ${\mathrm{p}}_{\mathrm{k}}$ such that $\mathrm{i}+\mathrm{j}+\mathrm{k}\ne 15$, is:

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is

Consider a rectangle $ABCD$ having $5,7,6,9$ points in the interior of the segments $AB,CD,BC,DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these point …

If the sides $AB,BC$ and $CA$ of triangle $ABC$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

The sum of all the 4-digit distinct numbers that can be formed with the digits $1,2,2$, and 3 is:

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is ____.

Total number of 6 - digit numbers in which only and all the five digits 1 , 3 , 5 , 7 and 9 appear, is

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that the most three of them are red is

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is:

If the number of five digit numbers with distinct digits and 2 at the $10^{\text{th}}$ place is 336 k , then k is equal to:

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

The value of $(2\cdot {}^1{P}_0-3\cdot {}^2{P}_1+4\cdot {}^3{P}_2-\dots$ up to $51^{\text{th}}$ term$)+(1!-2!+3!-\dots$ upt to $51^{\text{th}}$ term) is equal to:

The total number of 3 - digit numbers, whose sum of digits is 10 , is ______.

The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is ____.

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is ____.

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is

All possible numbers are formed using the digits $1,1,2,2,2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is:

The sum of all natural numbers '$n$' such that $100<n<200$ and H.C.F$(91,n)>1$ is:

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits $0,1,2,3,4,5$ (repetition of digits is allowed) is:

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:

The number of functions from $\{1,2,3,\dots ,20\}$ onto $\{1,2,3,\dots ,20\}$ such that $f(k)$ is a multiple of 3, whenever k is a multiple of 4, is:

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is:

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is

Consider three boxes, each containing 10 balls labelled $1,2,\dots \dots ,10$. Suppose one ball is randomly drawn from each of the boxes. Denote by ${\mathrm{n}}_{\mathrm{i}}$, the label of the ball drawn from the $i^{\text{th }}$ box, $(i=1,2,3)$. Then, the number of w …

Let $S=\{1,2,3,\dots ,100\}$. The number of non-empty subsets A of S such that the product of elements in A is even is:

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to:

There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84 , then the value of …

If ${}^n{\mathrm{C}}_4,{}^n{\mathrm{C}}_5$ and ${}^n{\mathrm{C}}_6$ are in A.P., then n can be :

The number of natural numbers less than 7,000 which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to:

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is:

The number of 6 digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by 11 and no digit is repeated, is

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

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 …

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is:

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0,0),(0,41)$ and $(41,0)$, is :

The number of integers greater than 6,000 that can be formed, using the digits $3,5,6,7$ and 8 , without repetition, is :

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is