ILL 1978 Problem 40

If $C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)$ , prove the identity $$C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}$$ and then evaluate the sum $$S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.$$