If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

If $(1 -x + x^2)^n = a_0 + a_1x + a_2x^2 + ……. + a_{2n}x^{2n}$, then $a_0 + a_2 + a_4 +……..+ a_{2n}$ is equal to

The sum to $(n + 1)$ terms of the series $\frac{{{C_0}}}{2} – \frac{{{C_1}}}{3} + \frac{{{C_2}}}{4} – \frac{{{C_3}}}{5} + …$ is 

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 ${}^{21}{C_1} + 3.{}^{21}{C_3} + 5.{}^{21}{C_5} + ……19{}^{21}{C_{19}} + 21.{}^{21}{C_{21}} = k$ Then number of prime factors of $k$ is