In the expansion of

$(2x + 1).(2x + 5) . (2x + 9) . (2x + 13)...(2x + 49),$ find the coefficient of $x^{12}$ is :-

The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:

Let $\alpha=\sum_{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$. If $5 \alpha=6 \beta$, then $n$ equals

If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) ...... (1 + x + x^2 + x^3 + ...... + x^n)$

$\equiv  a_0 + a_1x + a_2x^2 + a_3x^3 + ...... + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to

If $\sum_{ k =1}^{10} K ^{2}\left(10_{ C _{ K }}\right)^{2}=22000 L$, then $L$ is equal to $.....$

The sum of the coefficients in the expansion of ${(1 + x - 3{x^2})^{2163}}$ will be