The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is $………..$.

If all the six digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the $72^{\text {th }}$ number is $…………$.

In how many ways can the letters of the word $\mathrm{ASSASSINATION} $ be arranged so that all the $\mathrm{S}$ 's are together?

A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to

If $^{n} C _{9}=\,\,^{n} C _{8},$ find $^{n} C _{17}$