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

Words of length $10$ are formed by using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated and $y$ be the number of such words where exactly two letters are repeated twice and no other letter is repeated,then the value of $\frac{y}{x}$ is

There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

Let $n \geq 3$ be an integer. For a permutation $\sigma = (a_1, a_2, \ldots, a_n)$ of $(1, 2, \ldots, n)$,we define $f_\sigma(x) = a_n x^{n-1} + a_{n-1} x^{n-2} + \ldots + a_2 x + a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x) = 0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1, 2, \ldots, n)$ of the values $S_\sigma$. Then,

The number of positive integral solutions of the equation $xyz = 3000$ is

The number of four-digit numbers formed by using the digits $0, 2, 4, 5$ (without repetition) which are not divisible by $5$ is: