Sum of co-efficients of terms of degree $m$  in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is

If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}, p, q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^{3}$ is equal to $............$

The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta  + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^{n}-b^{n}$, whenever $n$ is a positive integer.

The coefficients of three consecutive terms in the expansion of $(1+a)^{n}$ are in the ratio $1: 7: 42 .$ Find $n$

In the expansion of ${(1 + x)^n}$ the coefficient of $p^{th}$ and ${(p + 1)^{th}}$ terms are respectively $p$ and $q$. Then $p + q = $