If ${ } ^ { 20 } \mathrm { C } _ { \mathrm { r } }$ is the co-efficient of $x ^ { \mathrm { r } }$ in the expansion of $( 1 + x ) ^ { 20 }$, then the value of $\sum _ { \mathrm { r } = 0 } ^ { 20 } \mathrm { r } ^ { 2 } \left( { } ^ { 20 } \mathrm { C } _ { \mathrm { r } } \right)$ is equal to:
(1) $420 \times 2 ^ { 18 }$
(2) $380 \times 2 ^ { 18 }$
(3) $380 \times 2^{19}$
(4) $420 \times 2 ^ { 19 }$