Explanation

(a)
If $A$ and $B$ are two sets having $m$ and $n$ elements respectively such that $1 \leq n \leq m$, then number of onto mapping from $A$ to $B$
$\ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{n}{( - 1)^{n - 1}nC_{r}r^{m}}$
Here, $\ \ m = 100$, $n = 2$
$\therefore$ The number of onto mappings from $A$ to $B$
$\ \ \ \ \ \ \ \ \ \ \ = \sum_{r = 1}^{2}{( - 1)^{2 - r}\ {\ ^{2}C}_{r}\ r^{100}}$
$\ \ \ \ \ \ \ \ \ \ = ( - 1)^{2 - 1.2}C_{1} \times 1^{100} + ( - 1)^{2 - 2.2}C_{2}.2^{100}$
$\ \ \ \ \ \ \ \ \ \ = 2^{100} - 2$