Compute $\frac{12!}{10! \times 2!}$

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

Out of $7$ consonants and $4$ vowels,the number of words consisting of $3$ consonants and $2$ vowels that can be formed is:

Determine $n$ if $^{2n}C_{3} : ^{n}C_{3} = 12 : 1$.

If $\binom{n}{3} + \binom{n}{4} > \binom{n+1}{3}$,then:

If $(2 \leq r \leq n)$, then ${}^{n}C_{r} + 2 \cdot {}^{n}C_{r+1} + {}^{n}C_{r+2}$ is equal to: