Evaluate $\frac{n!}{r!(n-r)!}$ when $n=5$ and $r=2$.

The value of ${ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4$ is

$A$ candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$, each containing $6$ questions. The candidate is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make his/her choice of $6$ questions?

$A$ question paper consists of two sections having $3$ and $4$ questions respectively. The following note is given on the paper: "It is not necessary to attempt all the questions. One question from each section is compulsory". In how many ways can a candidate select the questions?

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

In a library,there are $n$ distinct books and each has $p$ copies. What is the number of ways to select one or more books?