Permutations & Combinations Questions (855)

If all the six digit numbers x₁x₂x₃x₄x₅x₆ with 0 < x₁ < x₂ < x₃ < x₄ < x₅ < x₆ are arranged in the increasing order, then the sum of the digits in the 72nd number is _____.

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

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

It $5$ letters are put in the $5$ envelopes. Find the no. of ways so that atleast $2$ letters are in wrong envelope:

The largest value of $n$, for which $40^n$ divides $60!$, is

Number of ways in which 7 green bottles and 8 blue tube bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour):

You are given an unlimited supply of each of the digits $1, 2, 3$ or $4$. Using only these four digits, you construct $n$ digit numbers. Such $n$ digit numbers will be called LEGITIMATE if it contains the digit $1$ either an even number times or not at all. Number of $n$ digit legitimate numbers are:

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is

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

Number of ways in which four different toys and five indistinguishable marbles can be distributed between Amir, Aniket and Anthony, if each child receives at least one toy and one marble, is:

In $k$ ways can you place 2 rooks on a chessboard such that they are not in attacking positions, if rooks can attack only in a same row or in a same column? Then $\frac{k}{100}$ is_____.

The number 916238457 is an example of nine digit number which contains each of the digit 1 to 9 exactly once. It also has the property that the digits 1 to 5 occur in their natural order, while the digits 1 to 6 do not. Number of such numbers are:

There are $(p + q)$ different books on different topics in Mathematics. $(p \neq q)$ If $L =$ The number of ways in which these books are distributed between two students $X$ and $Y$ such that $X$ get $p$ books and $Y$ gets $q$ books. $M =$ The number of ways in which these books are distributed between two student $X$ and $Y$ such that one of them gets $p$ books and another gets $q$ books. $N =$ The number of ways in which these books are divided into groups of $p$ books and $q$ books then:

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 ordered pairs $(m, n), m, n \in \{1,2,....50\}$ such that $6^m + 9^n$ is a multiple of 5:

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

How many 5-digit numbers can be made from the digits 1, 2, 3, 4 so that all the digits are taken in each number?

How many ways are there to arrange the letters of the word "GARDEN" with the vowels in alphabetical order?

The number of ways in which 4 boys and 4 girls can stand in a circle so that each boy and each girl is one after the other is :

If $\binom{40}{0}+\binom{41}{1}+\binom{42}{2}+\cdots+\binom{60}{20}=\dfrac{m}{n}\binom{60}{20}$, $\gcd(m,n)=1$, max value of $m+n$ 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: (JEE Main 2019, April)

Let A1, A2, A3 ,............. An are the vertices of a polygon of sides n. If the pentagons that can be constructed by joining these vertices such that none of the side of the polygon is also the side of the pentagon is 36, then the value of n is equal to:

Number of natural numbers with 5 distinct digits formed using $\{1,2,3,5,6,7,9\}$ and divisible by 6 are

Four-digit numbers are formed using digits from $\{0,1,2,3,4,5\}$, repetition allowed. Statement $S_1$: the number of such odd numbers is 480. Statement $S_2$: the number formed with exactly three different digits is 360.

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

The number of straight lines that can be formed by joining 20 points of which 4 points are collinear is

A book has 618 pages. To number the pages, how many times a typist has to press keys if every page is to be numbered manually?

There are 4 girls and 6 boys to be seated such that no two girls sit together. In how many ways can it be done?

The number of four-letter words that can be formed using the letters of the word BARRACK is

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:

A committee of 12 is to be formed from 9 women and 8 men in which at least 5 women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively,

In how many different ways may 12 things, 4 each of three varieties, be distributed equally among two persons?

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

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose the chairs from amongst the chairs marked 1 to 4 and then the men select the chairs from amongst the remaining. The number of possible arrangements 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

Consider all possible permutations of the letters of the word ENDEANOEL. The number of permutations in which letters A, E, O occur only in odd positions, is

Let $A=\{1,2,3,\ldots,10\}$, $B=\{4,8,12,16,20\}$ and $C=\{D:D\subseteq A,\,D\cap B\neq\emptyset\}$. The number of elements in $C$ which have at least 3 but at most 6 cardinal number is

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

How many four digit numbers can be made from the digits 2, 3, 4, 5 and 6 without repeating any digit such that they are exactly divisible by 4?

The number of ways in which 5 beads of different colours form a necklace is

If $\binom{40}{0}+\binom{41}{1}+\binom{42}{2}+\cdots+\binom{60}{20}=\dfrac{m}{n}\binom{60}{20}$, $\gcd(m,n)=1$, max value of $m+n$ is

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

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

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is

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 ______.

If \(T_{n+1} - T_n = 10\), then the value of \(n\) is obtained as follows: \({}^{n+1}C_3 - {}^nC_3 = 10 \Rightarrow \dfrac{(n+1)n(n-1)}{6} - \dfrac{n(n-1)(n-2)}{6} = 10 \Rightarrow n(n-1)(n+1-n+2) = 60 \Rightarrow n(n-1) = 20 \Rightarrow n(n-1) = 5 \times 4\). Therefore, \(n =\):

How many 4-letter words can be made from the word MATHEMATICS such that no letter is repeated?

Find the highest power of 3 in \binom{50}{10}\.

There are 5 eligible Punjabi grooms of which 3 know Bengali and 5 eligible Bengali grooms of which 2 know Punjabi. There are 5 eligible Punjabi brides and 5 eligible Bengali brides. If an eligible groom is agreeable to marry a girl of his community or knowing her language and brides have no choice, in how many different ways 10 couples can be formed?

In a network of railways, a small island has 15 stations. Find the number of different types of tickets to be printed for each class, if every station must have tickets for other stations.