In how many ways can $35$ apples be distributed among $3$ boys such that each boy can have any number of apples?

If a committee of $10$ members is to be formed from $8$ men and $6$ women,then the number of different possible committees in which the men are in majority is

The value of ${ }^{16} C_{9}+{ }^{16} C_{10}-{ }^{16} C_{6}-{ }^{16} C_{7}$ is

$A$ candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$, each containing $6$ questions. The candidate is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make his/her choice of $6$ questions?

If $^nC_4$, $^nC_5$ and $^nC_6$ are in arithmetic progression ($A$.$P$.), then the value of $n$ is...

In how many ways can $11$ identical pencils be distributed among $6$ children such that each child receives at least one pencil?