In how many ways can one select a cricket team of $11$ from $17$ players in which only $5$ players can bowl,if each cricket team of $11$ must include exactly $4$ bowlers?

In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

Six married couples decide to form a committee of $6$ persons. The total number of ways to form the committee such that it contains no couples is equal to:

The domain of definition of the function $f(x) = \log_{[x + \frac{1}{x}]} |x^2 - x - 6| + ^{16-x}C_{2x-1} + ^{20-3x}P_{2x-5}$ is,where $[x]$ denotes the greatest integer function.

There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included,and $G$ and $H$ must not be included in the team of $5$?