IMO 1978 LL TUR40

Origin: TUR

Problem

If Cp n = n! p!(n−p)! (p \geq1), prove the identity Cp n = Cp−1 n−1 + Cp−1 n−2 + \cdot \cdot \cdot + Cp−1 p

• Cp−1 p−1 and then evaluate the sum S = 1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdot \cdot \cdot + 97 \cdot 98 \cdot 99.