The last two digits of $2015! + 3^{2015}$ are:

The total number of $5$-digit numbers, formed by using the digits $1, 2, 3, 5, 6, 7$ without repetition, which are multiples of $6$, is

$^n{P_r} \div ^n{C_r} = $

There are three sections in a question paper,each containing $4$ questions. If a candidate has to answer only $5$ questions from this paper without leaving any section,then the number of ways the candidate can make the choice of questions is:

The value of $(2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \dots \text{ up to } 51^{\text{th}} \text{ term}) + (1! - 2! + 3! - \dots \text{ up to } 51^{\text{th}} \text{ term})$ is equal to

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