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

Let $S=\{1,2,3,5,7,10,11\}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of $3$ , is $........$

In how many ways a team of $10$ players out of $22$ players can be made if $6$ particular players are always to be included and $4$ particular players are always excluded

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

The number of four letter words that can be formed using the letters of the word $BARRACK$ is

The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to