In the expansion of ${(1 + x)^n}$ the sum of coefficients of odd powers of $x$ is

If  $\sum\limits_{K = 1}^{12} {12K{.^{12}}{C_K}{.^{11}}{C_{K – 1}}} $ is equal to $\frac{{12 \times 21 \times 19 \times 17 \times …….. \times 3}}{{11!}} \times {2^{12}} \times p$ then $p$ is

If $C_r= ^{100}{C_r}$ , then $1.C^2_0 – 2.C^2_1 + 3.C^2_3 – 4.C^2_0 + 5.C^2_4 – …. + 101.C^2_{100}$ is equal to

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

In the expansion of ${(x + a)^n}$, the sum of odd terms is $P$ and sum of even terms is $Q$, then the value of $({P^2} – {Q^2})$ will be