If $\left({ }^{40} \mathrm{C}_0\right)+\left({ }^{41} \mathrm{C}_1\right)+\left({ }^{42} \mathrm{C}_2\right)+\ldots .+\left({ }^{60} \mathrm{C}_{20}\right)=\frac{m}{n}{ }^{60} \mathrm{C}_{20}, m$ and $n$ are coprime, then $m+n$ is equal to $\_\_\_\_$

Let $n \geq 5$ be an integer. If $9^n-8 n-1=64 \alpha$ and $6^n-5 n-1=25 \beta$, then $\alpha-\beta$ is equal to :

The remainder on dividing $1+3+3^2+3^3+\ldots+3^{2021}$ by 50 is $\_\_\_\_$

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $\left(2 x^3+\frac{3}{x}\right)^{10}$ is $5^{10}-\beta \cdot 3^9$...

The term independent of $x$ in the expression of $\left(1-x^2+3 x^3\right)\left(\frac{5}{2} x^3-\frac{1}{5 x^2}\right)^{11}, x \neq 0$ is

If the constant term in the expansion of $\left(3 x^3-2 x^2+\frac{5}{x^5}\right)^{10}$ is $2^k . l$, where $l$ is an odd integer, then the value of...

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\ldots . . x^{500} \ldots>0$, is

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2 x^3+\frac{3}{x^k}\right)^{12}, x \neq 0$ is $2^8 \cdot \ell$,...

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$ is $5^k l$, where $l, k \in N$ and $l$ is co- prime to...

If $\sum_{k=1}^{31}\left({ }^{31} C_k\right)\left({ }^{31} C_{k-1}\right)-\sum_{k=1}^{30}\left({ }^{30} C_k\right)\left({ }^{30} C_{k-1}\right)=\frac{\alpha(60!)}{(30!)(31!)}$, Where $\alpha \in R$, then the value of $16 \alpha$ is equal to