The value of $$20_{C_1} + 2 \times 20_{C_2} + 3 \times 20_{C_3} + ..... + 20 \times 20_{C_{20}}$$ is

Let S= $$\left(0\times\ ^{20}C_0\right)+\left(1\times^{20}C_1\ \right)+\left(2\times^{20}C_2\ \right)+\left(3\times^{20}C_3\ \right)+...\left(19\times^{20}C_{19}\ \right)+\left(20\times^{20}C_{20}\ \right)$$.---------Eqn 1.

When we reverse the RHS and write again, we get

S= $$\left(20\times\ ^{20}C_20\right)+\left(19\times^{20}C_19\ \right)+\left(18\times^{20}C_18\ \right)+\left(17\times^{20}C_17\ \right)+...\left(1\times^{20}C_{1}\ \right)+\left(0\times^{20}C_{0}\ \right)$$.--------- Eqn 2.

When we use $$^nC_r=^nC_{n-r}$$, we can re-write the second equation as:

S= $$\left(20\times\ ^{20}C_0\right)+\left(19\times^{20}C_1\ \right)+\left(18\times^{20}C_2\ \right)+\left(17\times^{20}C_3\ \right)+...\left(1\times^{20}C_{19}\ \right)+\left(0\times^{20}C_{20}\ \right)$$.-----------Eqn 3.

Adding Eqn 1 and Eqn 3, we will get,

2S= 20($$^{20}C_0+^{20}C_1+^{20}C_2+^{20}C_3.....+^{20}C_{19}+^{20}C_{20}$$

=> 2S=20$$\times\ 2^{20}$$

=>S= $$\ \frac{\ 20\times\ 2^{20\ }}{2}$$

.'. S= 20$$\times2^{19}\ $$