If the coefficient of the middle term in the expansion of ${(1 + x)^{2n + 2}}$ is $p$ and the coefficients of middle terms in the expansion of ${(1 + x)^{2n + 1}}$ are $q$ and $r$, then
$p + q = r$
$p + r = q$
$p = q + r$
$p + q + r = 0$
The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2 x^{3}+\frac{3}{x^{k}}\right)^{12}, x \neq 0$ is $2^{8} \cdot \ell$, where $\ell$ is an odd integer, is......
Coefficient of ${x^2}$ in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^8}$ is
The coefficient of ${x^5}$ in the expansion of ${(x + 3)^6}$ is
Find an approximation of $(0.99)^{5}$ using the first three terms of its expansion.
The coefficient of $x^{1012}$ in the expansion of ${\left( {1 + {x^n} + {x^{253}}} \right)^{10}}$ , (where $n \leq 22$ is any positive integer), is