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?

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

There are $25$ candidates for $12$ positions,out of which $5$ are from the reserved category. If $3$ positions are reserved and the remaining are open,in how many ways can the selection be made?

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

There are two urns. Urn $A$ has $3$ distinct red balls and urn $B$ has $9$ distinct blue balls. From each urn,two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

The number of $4$-digit numbers without repetition that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ in which each number has two odd digits and two even digits is: