Permutations and Combinations — JEE Main PYQs

The number of ways of getting a sum 16 on throwing a dice four times is ____.

Let the set $S=\{2,4,8,16,\dots ,512\}$ be partitioned into 3 sets $A,B,C$ with equal number of elements such that $A\cup B\cup C=S$ and $A\cap B=B\cap C=A\cap C=\varphi$. The maximum number of such possible partitions of $S$ is equal to

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $0\le r\le n$ If ${}^{n+1}{C}_{r+1}:{}^n{C}_r:{}^{n-1}{C}_{r-1}=55:35:21$, then $2n+5r$ is equal to

If all words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text{th}}$ position in this arrangement is

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some $\mathrm{m},\mathrm{n};{}^6{\mathrm{C}}_{\mathrm{m}}+2\left({}^6{\mathrm{C}}_{\mathrm{m}+1}\right)+{}^6{\mathrm{C}}_{\mathrm{m}+2}>{}^8{\mathrm{C}}_3$ and ${}^{n-1}{P}_3:{}^n{P}_4=1:8$, then ${}^n{P}_{m+1}+{}^{n+1}{C}_m$ is equal to

Let $\alpha =\frac{(4!)!}{(4!{)}^{3!}}$ and $\beta =\frac{(5!)!}{(5!{)}^{4!}}$. Then :

The lines ${\mathrm{L}}_1,{\mathrm{L}}_2,\dots ,{\mathrm{L}}_{20}$ are distinct. For $\mathrm{n}=1,2,3,\dots ,10$ all the lines ${\mathrm{L}}_{2\mathrm{n}-1}$ are parallel to each other and all the lines ${\mathrm{L}}_{2\mathrm{n}}$ pass through a given point P . The maximum n …

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

The number of elements in the set
$S=\{(x,y,z):x,y,z\in \mathbf{Z},x+2y+3z=42,x,y,z\ge 0\}$ equals ____.

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

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $\mathrm{A}$, $\mathrm{B}$ and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A …

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ____.

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is

The number of arrangements of the letter of the word "INDEPENDENCE" in which all the vowels always occur together is

The largest natural number $n$ such that $3^n$ divides 66! is ______.

If ${}^{2\mathrm{n}}{\mathrm{C}}_3:{}^{\mathrm{n}}{\mathrm{C}}_3=10:1$, then the ratio
$\left(n^2+3n\right):\left(n^2-3n+4\right)$ is

The number of ways of giving 20 distinct oranges to 3 children such that each child gets atleast one orange is ____

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $(6!)\mathrm{k}$, then k is equal to

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is:

The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to ______.

The number of permutations, of the digits $1,2,3$, ..... 7 without repetition, which neither contain the string 153 nor the string 2467, is ______.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840 , then the total numbers of persons, who participated in the tournament, is ____.

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits $0,1,3,5,7$ and 9 without repetition, is equal to

Let the digits $\mathrm{a},\mathrm{b},\mathrm{c}$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ______.

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.

The total number of three-digit numbers, divisible by 3, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is:

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct cod …

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

The number of triplets $(x,y,z)$, where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is

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

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ____.

The total number of six digit numbers, formed using the digits $4,5,9$ only and divisible by 6, is ______.

Number of integral solutions to the equation $\mathrm{x}+\mathrm{y}+\mathrm{z}=21$, where $\mathrm{x}\ge 1,\mathrm{y}\ge 3,\mathrm{z}\ge 4$, is equal to ______.

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is

If all the six digit numbers ${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{x}}_4{\mathrm{x}}_5{\mathrm{x}}_6$ with $0<{\mathrm{x}}_1<{\mathrm{x}}_2<{\mathrm{x}}_3<{\mathrm{x}}_4<{\mathrm{x}}_5<{\mathrm{x}}_6$ are arranged in the increasing order, then the sum of the di …

Five digit numbers are formed using the digits 1,2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____.