Bài toán:

Tính tổng $S = \,\,\frac{{{1^2}}}{2}C_{100}^1 + \frac{{{2^2}}}{3}C_{100}^2 + \frac{{{3^2}}}{4}C_{100}^3 + \cdots + \frac{{{{100}^2}}}{{101}}C_{100}^{100}$