The value of $\sum\limits_{r = 1}^{15} {{r^2} \left( \frac{^{15}C_r}{^{15}C_{r - 1}} \right)}$ is equal to

Let $P_m$ stand for $^nP_m$. Then the expression $1 \cdot P_1 + 2 \cdot P_2 + 3 \cdot P_3 + \dots + n \cdot P_n =$

If $S_n = \sum_{r=0}^n \frac{1}{^nC_r}$ and $T_n = \sum_{r=0}^n \frac{r}{^nC_r}$,then $\frac{T_n}{S_n}$ is equal to

If all possible $4$-digit numbers are formed by choosing $4$ different digits from the given digits $1, 2, 3, 5, 8$,then the sum of all such $4$-digit numbers is:

The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2, 3, 4, 5, 6$ (repetition of digits is not allowed) and divisible by $55$ is $....$

Four notes of Rs. $100$ and one note each of Rs. $1$,Rs. $2$,Rs. $5$,Rs. $20$,and Rs. $50$ are to be distributed among $3$ children such that each child receives at least one note of Rs. $100$. In how many ways can this distribution be done?