Permutation Combination

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

For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in \mathrm{S}_6$, but $\{1,2,4\} \notin \mathrm{S}_6$. Then…

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

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…

There are ten boys $B_{1},B_{2},\ldots\,.,B_{10}$and five girls $G_{1},G_{2},\ldots\,.G_{5}$in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_{1}$and…

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…

In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?

There are 10 points in a plane. No three of these points are in a straight line. What is the total number of straight lines which can be formed by joining the points?

What is the number of positive integer solutions of $x + y + z = 5$?

What is the number of different messages that can be represented by three 0's and two 1's?

What is the number of selections of at most 3 things from 6 different things?