Doubt on a math question

Question

Chk this question https://gateoverflow.in/100202/test-series-counting

$1)$Can someone verify this ans??

See if $\left ( _{0}^{6}\textrm{C} \right )$ in one set, other set will contain $\left ( _{6}^{6}\textrm{C} \right )$ elements.

right??

Now why do we again need $2^{n}$ arrangement?? I mean what is meaning of it??

$2)$ How $\sum_{I=0}^{n}\left ( _{i}^{n}\textrm{C} \right ).2^{n-i}=3^{n}$??