For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. If $S$ denotes the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find $S$.

Show that $$\sum_{i=0}^n \binom{n}{i}^2 = \binom{2n}{n}$$.

Simplify the expression $$\binom{2020}{0}^2 + \binom{2020}{1}^2 + \cdots + \binom{2020}{2020}^2$$