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2018/19 AHSMC Part 1 Problem 7

Problem 7 of 16

How many triplets (m,n,k)(m,n,k), of integers greater than one, exist such that (m!)⋅(n!)=(k!)(m!) \cdot (n!) = (k!)? (Notice that n!=1×2×3×⋯×nn! = 1 \times 2 \times 3 \times \cdots \times n.)

Answer choices

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Solution

Let nn be any positive integer greater than 22. Setting m=n!−1m = n! - 1 and k=n!k = n! one obtains (n!−1)!⋅(n!)=(n!)!(n! - 1)! \cdot (n!) = (n!)! and hence the equation has infinitely many solutions. The answer is (E).

University of Alberta