$^nC_r + 2^nC_{r-1} + ^nC_{r-2} = $

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

In how many ways can a team of $10$ players be selected from $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded?

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him is

Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

The total number of ways of selecting six coins out of $20$ one rupee coins,$10$ fifty paise coins,and $7$ twenty-five paise coins is: