The coefficient of $x^{13}$ in the expansion of $(1 - x)^5(1 + x + x^2 + x^3)^4$ is :-

If $a_r$ is the coefficient of $x^r$ in the expansion of $(1 + x + x^2)^n$,then $a_1 - 2a_2 + 3a_3 - \dots - 2n\,a_{2n} = $

The value of $\sum\limits_{n = 0}^4 {{{\left( {1009 - 2n} \right)}^4} \binom{4}{n} {\left( { - 1} \right)^n}}$ is

If $C_r$ denotes the binomial coefficient ${ }^{n} C_r$,then $(-1) C_0^2+2 C_1^2+5 C_2^2+\ldots+(3 n-1) C_n^2$ is equal to

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:

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 $1-\frac{F}{1+(\sqrt{2}-1)^4}=$