Out of $6$ books,in how many ways can a set of one or more books be chosen?

$A$ committee of $12$ is to be formed from $9$ women and $8$ men such that at least $5$ women are included. Find the number of committees in which women are in the majority and the number of committees in which men are in the majority,respectively.

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables such that at least one officer is included?

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

In how many ways can we select any $4$ letters from the word $CORGOO$?