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

If ${ }^9 C_3+{ }^9 C_5={ }^{10} C_r$ for some $r \in N$,then $r=$

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq -1, w \geq 1$ is

Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

If $20$ identical books are to be distributed among $4$ persons such that each person receives at least one book,in how many ways can this be done?