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

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

The number of ways in which any four letters can be selected from the word '$CORGOO$' is

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals

In how many ways can $10$ balls be divided between two boys,such that one receives $2$ balls and the other receives $8$ balls?

The least value of a natural number $n$ such that $\binom{n-1}{5} + \binom{n-1}{6} < \binom{n}{7}$ is: