In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

If ${ }^{11} C_4+{ }^{11} C_5+{ }^{12} C_6+{ }^{13} C_7={ }^{14} C_{r}$,then the value of $r$ is

$A$ bag contains $5$ black and $6$ red balls. Determine the number of ways in which $2$ black and $3$ red balls can be selected.

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is