The number of ways in which $16$ identical oranges can be distributed among $4$ children such that each child gets at least one orange is:

The total number of ways of forming a committee of $5$ members out of $7$ Indians,$6$ Americans,$5$ Russians,and $4$ Australians such that every committee contains at least one member from each country is:

$A$ committee of $4$ members is to be formed from $4$ couples (husbands and wives). In how many ways can the committee be formed such that no couple is included in the committee?

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:

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

The value of $^{15}C_{3} + ^{15}C_{13}$ is