Competition problem — Combinatorics

Library / /36 of 520

Combinatorics Difficulty 4.9 AIME Find the answer

There are 20 people - 10 boys and 10 girls. How many ways are there to form a company where the number of boys and girls is equal?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

For each such company, associate a set of 10 people, which includes all the girls who joined the company and all the boys who did not join it.

## Solution

Let there be some company consisting of $k$ boys and $k$ girls. We will associate it with a set of 10 people, which includes $k$ girls who joined the company and $10-k$ boys who did not join it.

The established correspondence is clearly one-to-one. Therefore, the desired number is equal to the number of ways to choose 10 people out of 20.

## Answer

$C_{20}^{10}=184756$ ways.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.