Permutations & Combinations Questions (855)

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

There are red, green and white identical balls, each being 10 in number. The number of selections of 10 balls in which the number of red balls is double the number of green balls is ______.

To fill 8 vacancies there are 24 candidates of which 6 are from scheduled castes, 5 are from other backward castes. If 25% and \(12\frac{1}{2}\%\) of the vacancies are reserved for the scheduled castes and other backward castes respectively while the rest are open to all, find the number of ways to make the selection of candidates.

The number of ways to choose a subset of two elements {a, b} from the set {1, 2, 3, 4, \ldots, 49, 50} such that |a - b| \leq 5 is

The sequence of all positive palindromes are written in ascending order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, ...Then the 2018th positive palindrome is

The letters of the word 'DELHI' are arranged in all possible ways as in a dictionary, the rank of the word 'DELHI' is

The number of selections of 6 different letters that can be made from the words SUMAN and DIVYA so that each selection contains 3 letters from each word is ___.

Example 94: Find the number of different selections of 5 letters which can be made from 5A's, 4B's, 3C's, 2D's and 1E.

In how many ways can 6 boys and 4 girls sit in a row so that no boy is between two girls?

A badminton club has 10 couples as members. They meet to organise a mixed double match. If each wife refuse to partner as well as oppose her husband in the match then in how many different ways can the match be arranged?

16 persons are to be seated in a row so that in half of the seats from one side 4 particular persons sit consecutively and in the other half 3 particular persons sit together. Find the number of ways in which they can be seated.

Rekha married Shivram and had 4 sons. Varsha married Ajoy and had 4 sons. Both the couples had divorce and after that Shivram married Varsha while Ajoy married Rekha. They too had 3 sons each from their wedlocks. How many selections of 8 children can be made from the 14 children so that each of them have equal number of sons in the selection?

Let $A=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\min\{\binom{2023}{i},\binom{2023}{j}\}$ and $B=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\max\{\binom{2023}{i},\binom{2023}{j}\}$. Then $A+B$ equals

A boat is to be manned by 9 crew with 4 on the stroke side, 4 on the row side and one to steer. There are 11 crew of which 2 can stroke only, 1 can row only while 3 can steer only. In how many ways the crew can be arranged for the boat?

The number of non-negative integral solutions of \(2x + y = 11\) is ______.

Find the sum of all natural numbers \(n\) such that \(1000 \leq n \leq 4000\), that can be made with the digits 0, 1, 2, 3, 4 if repetition of digits in the same number is allowed.

In how many ways can a word of five letters beginning with a capital letter and consisting of two vowels and two consonants be formed with the capital letters C, A, C; vowels o, u, i, e and the consonants b, l, p, t, s?

Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. The number of triangles that can be formed using these points is ___.

In how many ways can 10 different prizes be given to 5 students if one particular boy must get 4 prizes and rest of the students can get any number of prizes?

Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of different ways in which the sitting arrangement can be made.

The sum of the odd divisors of $10!$ is

There are tea cups with and without handles. The number of ways of selecting 2 without handle and 3 with handle is exactly 1200. What is the maximum possible number of cups in the kitchen?

A bag contains 4 one-rupee coins, 2 twenty-five-paisa coins and 5 ten-paisa coins. In how many ways can an amount not less than Re 1 be taken out from the bag? Consider coins of the same denomination to be identical.

In an Ice cream parlor at South City Mall Kolkata, 4 different varieties of ice creams namely Vanila, Strawberry, Chocolate and Butter Scotch were available. On a particular day it was noticed that each customer buys at least one ice cream and at max 10 ice creams, on further investigation it was noticed that no two customer buys same set of ice creams then find the number of customers visited the ice cream shop on that particular day.

How many numbers of seven digits can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that odd digits always occupy odd places?

The number of words that can be made with all the letters of the world BHARATI so that the order of consonants and the order of vowels do not change is ______.

Number of ways in which 12 different things can be distributed in 3 groups, is

Find the number of three digit numbers which can be formed using the digits 0, 1, 2, 2, 3, 3, 3.

Number of ways in which 12 different things can be divided among five persons so that they can get 2, 2, 2, 3, 3 things respectively, is

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 different ways in which five "alike dashes" and eight "alike dots" can be arranged using only seven of these "dashes" and "dots" is

238. From an unlimited number of red, white, blue and green balls, a selection of 18 balls is to be made so that there are at least two of each colour. If the number of selection is \(k\), then \(k\) is equal to:

Let $A = \{1, a_1, a_2, \ldots, a_{18}, 77\}$ be a set of integers with $1 < a_1 < a_2 < \ldots < a_{18} < 77$. Let $A+A = \{x+y : y\in A\}$ contain exactly 39 elements. Then $a_1+a_2+\ldots+a_{18}$ equals

Letters of MATHS permuted in dictionary order. Serial number of THAMS is

The number less than 1000 that can be formed using the digits 0, 1, 2, 3, 4, 5 when repetition is not allowed is equal to

Find the total number of nine-digit numbers that can be formed using the digits 2, 2, 3, 3, 5, 5, 8, 8, 8 so that the odd digits occupy the even places.

If n1 and n2 are five-digit numbers, find the total number of ways of forming n1 and n2 so that these numbers can be added without carrying at any stage.

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

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is

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

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

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

The number of three-digit numbers $\overline{abc}$ which satisfy $a \leq b > c$ is

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:

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 $B_{1}$ and $B_{2}$ are not adjacent to each other, 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:

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:

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:

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 \rule{2cm}{0.4pt}.

The letters of the word KANPUR are written in all possible orders, and these words are written in dictionary (alphabetical) order. The word that appears at the $440^{\text{th}}$ position is