The coefficient of $x^{50}$ in the expansion of $(1+x)^{101}\left(1-x+x^2\right)^{100}$ is

Numerically greatest term in the expansion of $(2 x-3 y)^{11}$ when $x=\frac{1}{3}$ and $y=\frac{1}{2}$ is

$\frac{1}{8}-\frac{7}{8 \cdot 12}+\frac{7 \cdot 10}{8 \cdot 12 \cdot 16}-\ldots=$

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