The coefficient of the middle term in the expansion of $(1 + x)^{10}$ is

The $11^{th}$ term in the expansion of $\left(x+\frac{1}{\sqrt{x}}\right)^{14}$ is

Let $s_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j}$,$s_2 = \sum_{j=1}^{10} j \binom{10}{j}$,and $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j}$.
Statement $-1$: $s_3 = 55 \times 2^9$
Statement $-2$: $s_1 = 90 \times 2^8$ and $s_2 = 10 \times 2^8$

The largest term in the expansion of $(3 + 2x)^{50}$ where $x = \frac{1}{5}$ is

The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt{3}}}{{{x^3}}}} \right)^{10}}$ is

The expansion of $(a+x)^n$ contains $15$ terms. When $x=1$,the ratio of the neighboring terms to the middle term in this expansion is $16$. Then the positive integral value of '$a$' is