Track / Stage 4 / 40 of 340 #300 of 1964

Problem 300

AMC 12 late, AIME early

Combinatorics Difficulty 4.7 Multiple choice

14. In the annual football league of a certain middle school, it is stipulated that each team must play one match against every other team. If in the 2012 season all teams played a total of 21 matches. Then the number of participating teams is:

Pick one

Official solution

14. B.

There are $n$ teams participating in the league.
$\text { Then } \frac{n(n-1)}{2}=21 \Rightarrow n=7 \text {. }$

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