Division of Objects into Groups

A class has $n$ students. We have to form a team of the students including at least two students and also excluding at least two students. The number of ways of forming the team is

A class has $21$ students. The class teacher has been asked to make $n$ groups of $r$ students each and go to zoo taking one group at a time. The size of group zoo i.e., the value of $r$) for which the teacher goes to the maximum number of times is (no group can go to the zoo twice)

$S=\left\{1,2,3,\dots ,20\right\}$ is to be partitioned into four sets $A, B, C$ and $D$ of equal size. The number of ways it can be done is

The number of ways to give $16$ different things to three persons, so that $B$ gets $1$ more than $A$ and $C$ gets $2$ more than $B$, is

The number of permutations of the letters of the word $\mathrm{HINDUSTAN}$ such that neither the pattern $\mathrm{HIN}$ nor $\mathrm{DUS}$ nor $\mathrm{TAN}$ appear, are

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. The number of ways in which we can place the balls in the boxes (order is not considered in the box ) so that no box remains empty is

If $n$ objects are arranged in a row, then the number of ways of selecting three of these objects, so that no two of them are next to each other, is

The number of non-negative integral solutions of $x+y+z\le n,$ where $n\in N$ is

The number of integral solutions of $x+y+z=0$ with $x\ge -5, y\ge -5, z\ge -5$ is

The number of integers which lie between $1$ and ${10}^6$ and which have the sum of the digits equal to $12,$ is

The number of ways in which $mn$ students can be distributed equally among $m$ sections, is

The number of integral solutions for the equation $x+y+z+t=20,$ where $x,y,z,t\ge -1,$ is

The number of ways of distributing $50$ identical things among $8$ persons in such a way that three of them get $8$ things each, two of them get $7$ things each and remaining $3$ get $4$ things each, is

The total number of ways of dividing $15$ different things into groups of $8,4$ and $3$ respectively, is