There are $m$ books in black cover and $n$ books in blue cover, and all books are different. The number of ways these $(m+n)$ books can be arranged on a shelf so that all the books in black cover are put side by side is

The number of ways in which an examiner can assign $30$ marks to $8$ questions, giving not less than $2$ marks to any question, is

If ${ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^{\mathrm{q}} \mathrm{P}_{\mathrm{r}}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1$ then ${ }^{\mathrm{q}+\mathrm{s}} \mathrm{C}_{\mathrm{r}-\mathrm{s}}$ is equal to .... .

The number of groups that can be made from $5$ different green balls, $4$ different blue balls and $3$ different red balls, if at least $1$ green and $1$ blue ball is to be included

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

A group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least one boy and one girl ?