The coefficient of $x y^{2} z^{3}$ in the expansion of $(x-2 y+3 z)^{3}$ is

The set of all real values of $x$ for which the expansion of $\left(125 x^{2}-\frac{27}{x}\right)^{\frac{-2}{3}}$ is valid, is

If $3^{2 n+2}-8 n-9$ is divisible by $2^{p}, \forall n \in \mathrm{~N}$, then the maximum value of $P$ is

If the coefficient fo $x^{r}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{100}$ is $a_{r}$ and $S=\sum_{r=0}^{300} a_{r}$ then $\sum_{r=0}^{300} r \cdot a_{r}=$

If $X \sim B(6, p)$ is a binomial variate and $\frac{P(X=4)}{P(X=2)}=\frac{1}{9}$, then $p=$

If $p$ and $q$ are the real numbers such that the 7 th term in the expansion of $\left(\frac{5}{p^3}-\frac{3 q}{7}\right)^8$ is 700 , then $49 p^2=$

If $T_4$ represents the 4 th term in the expansion of $\left(5 x+\frac{7}{x}\right)^{\frac{-3}{2}}$ and $x \notin\left[-\sqrt{\frac{7}{5}}, \sqrt{\frac{7}{5}}\right]$, then $\left(x^7 \sqrt{5 x}\right) T_4=$

If the coefficients of 3 consecutive terms in the expansion of $(1+x)^{23}$ are in arithmetic progression, then those terms are

The numerically greatest term in the expansion of $(3 x-16 y)^{15}$, when $x=\frac{2}{3}$ and $y=\frac{3}{2}$, is

For $n \in N$ the largest positive integer that divides $81^n+20 n-1$ is $k$. If $S$ is the sum of all positive divisors of $k$, then $S-k=$

If the term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then $k=$

The number of rational terms in the binomial expansion $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is

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