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Ways that the product of n distinct letters be formed?

Source: IMO Longlist 1989, Problem 44

September 18, 2008

combinatorics unsolvedcombinatorics

Problem Statement

Given two distinct numbers

b_1

and

b_2

, their product can be formed in two ways:

b_1 \times b_2

and

b_2 \times b_1.

Given three distinct numbers,

b_1, b_2, b_3,

their product can be formed in twelve ways:

b_1\times(b_2 \times b_3);

(b_1 \times b_2) \times b_3;

b_1 \times (b_3 \times b_2);

(b_1 \times b_3) \times b_2;

b_2 \times (b_1 \times b_3);

(b_2 \times b_1) \times b_3;

b_2 \times(b_3 \times b_1);

(b_2 \times b_3)\times b_1;

b_3 \times(b_1 \times b_2);

(b_3 \times b_1)\times b_2;

b_3 \times(b_2 \times b_1);

(b_3 \times b_2) \times b_1.

In how many ways can the product of

n

distinct letters be formed?

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