Given below are two statements :
Statement I: $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7 .
Statement II: The integral part of $(7+4 \sqrt{3})^{23}$ is an odd...

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots+x^{1000}$ is :

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^k}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

The sum of all possible values of $\mathrm{n} \in \mathbf{N}$, so that the coefficients of $x_2 x^2$ and $x^3$ in the expansion of $\left(1+x^2\right)^2(1+x)^{\mathrm{n}}$, are...

The value of $\frac{{ }^{100} C_{50}}{51}+\frac{{ }^{100} C_{51}}{52}+\ldots .+\frac{{ }^{100} C_{100}}{101}$ is:

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100}$ is equal to

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $(1+x)^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}, 0 \leq \mathrm{r} \leq \mathrm{n}$. If $...

If $\left(\frac{1}{{ }^{15} \mathrm{C}_0}+\frac{1}{{ }^{15} \mathrm{C}_1}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_1}+\frac{1}{{ }^{15} \mathrm{C}_2}\right) \ldots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)=\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_0{ }^{14} \mathrm{C}_1 \cdots{ }^{14} \mathrm{C}_{12}}$, then $30 \alpha$...

The largest $\mathrm{n} \in \mathbf{N}$, for which $7^{\mathrm{n}}$ divides $101!$, is :

If the coefficient of x in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$ and $x^3$ are both zero,...