The sum of the series 2 . 20 C 0 + 5 . 20 C 1 + 8 . 20 C 2 + 11 . 20 C 3 + . . . . . . . + 62 . 20 C 20 is…

Let,

$S= 2{.}^{20}{C}_0+5{.}^{20}{C}_1+8{.}^{20}{C}_3+.......+62{.}^{20}{C}_{20} ...\left(\mathrm{i}\right)$
$S=62{.}^{20}{C}_0+59{.}^{20}{C}_1+56{.}^{20}{C}_3+.......+2{.}^{20}{C}_{20}\mathrm{ } ...\left(\mathrm{ii}\right)$ (Writing in reverse order ${}^n{C}_r{=}^n{C}_{n-r}$ )
Adding $\left(i\right)$ and $\left(ii\right)$
$2S=64{(}^{20}{C}_0{+}^{20}{C}_1{+}^{20}{C}_2+.....{+}^{20}{C}_{20})$
$\Rightarrow \mathrm{S}={32.2}^{20}=2^{25}$