The value of $4 \{^nC_1 + 4 . ^nC_2 + 4^2 . ^nC_3 + ...... + 4^{n - 1}\}$ 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

${n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$ is

If $f(y) = 1 - (y - 1) + {(y - 1)^2} - {(y - 1)^{^3}} + ... - {(y - 1)^{17}},$ then the coefficient of $y^2$ in it is

The number $111......1 $ ( $ 91$ times) is

Total number of terms in the expansion of $\left[ {{{\left( {1 + x} \right)}^{100}} + {{\left( {1 + {x^2}} \right)}^{100}}{{\left( {1 + {x^3}} \right)}^{100}}} \right]$ is