$\sum_{\substack{i, j=0 \\ i \neq j}}^{n} {}^{n}C_{i} {}^{n}C_{j}$ is equal to

The coefficient of $x^{10}$ in the expansion of $(1 + x)^2 (1 + x^2)^3 (1 + x^3)^4$ is equal to

Let $a_1, a_2, a_3, \dots, a_{100}$ be positive real numbers and $S_k$ be the sum of products of $a_1, a_2, \dots, a_{100}$ taken $k$ at a time. If $S_{98} S_2 \ge \lambda (a_1 a_2 \dots a_{100})$,then $\lambda$ is

The sum of all the coefficients in the binomial expansion of $(1+2x)^n$ is $6561$. Let $R=(1+2x)^n=I+F$,where $I \in N$ and $0 < F < 1$. If $x=\frac{1}{\sqrt{2}}$,then $1-\frac{F}{1+(\sqrt{2}-1)^4}=$

For the natural numbers $m, n$,if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots +a_{m+n} y^{m+n}$ and $a_{1}=a_{2}=10$,then the value of $(m+n)$ is equal to:

If $(1 + x - 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + \dots$,then $a_0 - a_1 + a_2 - a_3 + \dots$ ends with: