Question

Let $1 + x = r$

$\therefore S = 1.r + 2.r^{2} + 3.r^{3} + \ldots\ldots + 100r^{100}\ldots\ldots$.

$$\begin{array}{r} rS = 1.r^{2} + 2.r^{3} + \ldots.. + 99r^{100} + 100r^{101}\#(AGP) \end{array}$$

(1) - (2) gives

$$S = - \frac{(1 + x)^{101}}{x^{2}} + \frac{1}{x^{2}} + \frac{100(1 + x)^{101}}{x} $$∴ coefficient $x^{48}$ in S

$= -$ coefficient of $x^{48}$ in $\frac{(1 + x)^{101}}{x^{2}} + 100$.

Coefficient of $x^{48}$ in $\frac{(1 + x)^{101}}{x}$

$$= 100\ ^{101}C_{49} - \ ^{101}C_{50}$$