IMO 1989 LL INA46

Origin: INA

Problem

Given two distinct numbers b1 and b2, their product can be formed in two ways: b1 \times b2 and b2 \times b1. Given three distinct numbers, b1, b2, b3, their product can be formed in twelve ways: b1 \times(b2 \timesb3); (b1 \times b2) \times b3; b1 \times (b3 \times b2); (b1 \times b3) \times b2; b2 \times (b1 \times b3); (b2 \times b1) \times b3; b2 \times (b3 \times b1); (b2 \times b3) \times b1; b3 \times (b1 \times b2); (b3 \times b1) \times b2; b3 \times (b2 \times b1); (b3 \times b2) \times b1. In how many ways can the product of n distinct letters be formed?