A  man $X$  has $7$  friends, $4$  of them are ladies and  $3$ are men. His wife $Y$ also has $7$ friends, $3$ of  them are  ladies and $4$ are men. Assume $X$ and $Y$ have no comman friends. Then the total number of ways in which $X$ and $Y$ together  can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is :

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

If $^{n}{P_4} = 24.{\,^n}{C_5},$ then the value of $n$ is

If $P(n,r) = 1680$ and $C(n,r) = 70$, then $69n + r! = $

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 $........$

A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of: 

exactly $3$ girls $?$