The value of ${ }^{16} C_{9}+{ }^{16} C_{10}-{ }^{16} C_{6}-{ }^{16} C_{7}$ is

There are $5$ apples,$4$ mangoes,$3$ oranges,and $1$ each of $2$ other varieties of fruits. The number of ways of selecting at least one fruit of each kind is

Five students are selected from $n$ students such that the ratio of the number of ways in which $2$ particular students are selected to the number of ways $2$ particular students are not selected is $2:3$. Then the value of $n$ is

$^nC_r + ^nC_{r-1}$ is equal to

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is

The number of integral solutions to the equation $x+y+z=21$,where $x \geq 1, y \geq 3, z \geq 4$,is equal to $..........$.