Explanation

(a)
We have,
$\sum_{r = 0}^{n}{\sum_{s = 0}^{n}{(r{+ s)C}_{r}C_{s} =}}\sum_{r = 0}^{n}{2rC_{r}^{2} + 2}\sum_{0 \leq r < s \leq n}^{}{\sum_{}^{}(r + s)}C_{r}C_{s}$
$\Rightarrow \sum_{r = 0}^{n}{\sum_{s = 0}^{n}{(r{+ s)C}_{r}C_{s} =}}2\sum_{r = 0}^{n}{r \bullet C_{r}^{2} + 2}\sum_{0 \leq r < s \leq n}^{}{\sum_{}^{}(r + s)}C_{r}C_{s}$
$\Rightarrow n \bullet 2^{2n} = 2 \bullet \left( \frac{n}{2}\ ^{2n}C_{n} \right) + 2\sum_{0 \leq r < s \leq n}^{}{\sum_{}^{}(r + s)}C_{r}C_{s}$
$\Rightarrow \sum_{0 \leq r < s \leq n}^{}{\sum_{}^{}(r + s)}C_{r}C_{s} = \frac{1}{2}\left\lbrack n \bullet 2^{2n} - n \bullet \ ^{2n}C_{n} \right\rbrack$
$= \frac{n}{2}\left\lbrack 2^{2n} - \frac{2n}{n}\ ^{2n - 1}C_{n - 1} \right\rbrack$
$= n\left\lbrack 2^{2n - 1} - \ ^{2n - 1}C_{n - 1} \right\rbrack$