If $1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k$, where $\mathrm{m}, \mathrm{n}, \mathrm{k} \in \mathrm{N}$, then...

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^n, n \leq 15$. If the sum of the Coefficients of the remaining terms...

The number of elements in the set $\left\{n \in N: 10 \leq n \leq 100\right.$ and $3^n-3$ is a multiple of 7$\}$ is

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Let $\left(a+b x+c x^2\right)^{10}=\sum_{i=0}^{20} p_i x^i, a, b, c \in \mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to

The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}: 1$,...

The mean of the coefficients of $x, x^2$ $\_\_\_\_$ $x^7$ in the binomial expansion of $(2+x)^9$ is $\_\_\_\_$ .

The largest natural number $n$ such that $3^n$ divides $66!$ is $\_\_\_\_$

Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal...