Competition problem — Combinatorics

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Combinatorics Difficulty 4.3 AIME Find the answer United States

Problem:

Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. (Please write your answer as an integer, without any incomplete computations.)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Answer: $100800$

Since there are $9$ cow and pigs combined and $7$ horses, there must be a pair with $1$ cow and $1$ pig, and all the other pairs must contain a horse. There are $4 \times 5$ ways of selecting the cow-pig pair, and $7!$ ways to select the partners for the horses. It follows that the answer is $4 \times 5 \times 7! = 100800$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.