Explanation

(b)
The number of ordered triples of positive integers which are solution of $x + y + \mathcal{z} = 100$
$=$coefficient of $x^{100}\ in\ \left( x + x^{2} + x^{3} + .. \right)^{3}$
=coefficient of $x^{100}in\ x^{3}(1 - x)^{- 3}$
=coefficient of $x^{97}$in
$\left( 1 + 3x + 6x^{2} + .\ldots\ldots.. + \frac{(n + 1)(n + 2)}{2}x^{n} + ... \right)$
$= \frac{(97 + 1)(97 + 2)}{2} = 49 \times 99 = 4851$