$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + …. + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n – 2}}{C_n}$ equals

If ${C_r}$ stands for $^n{C_r}$, the sum of the given series $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 – 2C_1^2 + 3C_2^2 – ….. + {( – 1)^n}(n + 1)C_n^2]$, Where $n$ is an even positive integer, is

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+$ $x^4(1+x)^{96}+\ldots \ldots . .+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_p-{ }^{46} \mathrm{C}_{\mathrm{q}}$.

Then a possible value to $\mathrm{p}+\mathrm{q}$ is :

If $\sum\limits_{r = 0}^{25} {\left\{ {^{50}{C_r}.{\,^{50 – r}}{C_{25 – r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} $, then $K$ is equal to