Official solution
S = \[{1^2}\,{}^{20}{C_1} + {2^2}\,{}^{20}{C_2} + {3^2}\,{}^{20}{C_3} + .......... + {20^2}\,{}^{20}{C_{20}}\]
\[ \Rightarrow \] \[\sum\limits_{r = 1}^{20} {{r^2}\,{}^{20}{C_r}} \]
\[ \Rightarrow \] \[\sum\limits_{r = 1}^{20} {r\,.\left( {r.{}^{20}{C_r}} \right)} \]
\[ \Rightarrow \] \[20\sum\limits_{r = 1}^{20} {r\,.} {}^{19}{C_{r - 1}}\]
\[ \Rightarrow \] \[20\sum\limits_{r = 1}^{20} {(r - 1 + 1)\,.} {}^{19}{C_{r - 1}}\]
\[ \Rightarrow \] \[20\sum\limits_{r = 1}^{20} {(r - 1)\,.} {}^{19}{C_{r - 1}} + 20\sum\limits_{r = 1}^{20} {{}^{19}{C_{r - 1}}} \]
\[ \Rightarrow \] \[20 \times 19\sum\limits_{r = 2}^{20} {{}^{18}{C_{r - 2}}} + 20 \times {2^{19}}\]
\[ \Rightarrow \] 20 \[ \times \] 19 \[ \times \] 218 + 20 \[ \times \] 219
\[ \Rightarrow \] \[20 \times {2^{18}}\left( {19 + 2} \right) = 20 \times 21 \times {2^{18}}\]
\[ \Rightarrow \] \[420 \times {2^{18}}\]