In an examination there are three multiple choice questions and each question has $4 $ choices. Number of ways in which a student can fail to get all answers correct, is

Let $S=\{1,2,3, \ldots ., 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_K$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

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

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 $............$.

A class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$, then $b +3\,g$ is equal to.

If $x,\;y$ and $r$ are positive integers, then $^x{C_r}{ + ^x}{C_{r - 1}}^y{C_1}{ + ^x}{C_{r - 2}}^y{C_2} + .......{ + ^y}{C_r} = $