How many $4$-digit numbers are there with distinct digits?

If $a_n = \sum_{r = 0}^n \frac{1}{^nC_r}$,then $\sum_{r = 0}^n \frac{r}{^nC_r}$ equals

If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon is

Given $6$ different letters of an alphabet. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together,is:

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first $2$ questions have $3$ choices each,the next $2$ have $4$ choices each,and the last $2$ have $5$ choices each?

$6$ boys and $5$ girls sit in a line such that $(I)$ no two girls sit together $(II)$ all the girls sit together. If $p$ is the number of arrangements in case $(I)$ and $q$ is the number of arrangements in case $(II)$,then $p/q =$