Binomial Theorem - Algebra - Mathematics Previous Year Questions

The constant term in the expansion of $\left(1+\frac{1}{x}\right)^{20}\left(30 x(1+x)^{29}+(1+x)^{30}\right)$ is

When $|x|>3$, then coefficient of $\frac{1}{x^n}$ in the expansion of $x^{3 / 2}(3+x)^{1 / 2}$ is

If $X \sim B(7, P)$ is a binomial variate and $P(X=3)=P(X=5)$, then $P=$

If the coefficient of 3rd term from the beginning in the expansion of $\left(a x^2-\frac{8}{b x}\right)^9$ is equal to the coefficient of 3rd term from the end in the expansion of $\left(a x-\frac{2}{b x^2}\right)^9$, then the relation between $a$ and $b$ is

If $C_0, C_1, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then the value of $\Sigma r^3 \cdot C_r$ when $n=5$ is

The coefficient of $x^{12}$ in the expansion of $\left(x^2+2 x+2\right)^8$ is

If $C_0, C_1, C_2, \ldots, C_8$ are the binomial coefficients in the expansion of $(1+x)^8$, then $\sum\limits_{r = 1}^8 {} r^3 \frac{C_r}{C_{r-1}}=$

Numerically greatest term in the expansion of $(2 x-3 y)^n$ when $x=\frac{7}{2}, y=\frac{3}{7}$ and $n=13$ is

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}$ is $\alpha_p$, then $\alpha_k-\alpha_{k+1}-\alpha_{k-1}=$

Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is