In how many ways can a committee be formed of $5$ members from $6$ men and $4$ women if the committee has at least one woman?

There are $n$ white and $n$ black balls marked $1, 2, 3, ..., n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours are

The number of ways in which $5$ boys and $3$ girls can be seated in a row so that each girl is between two boys is:

There are $n$ distinct points on the circumference of a circle. The number of pentagons that can be formed with these points as vertices is equal to the number of possible triangles. Then the value of $n$ is

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

$A$ 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