# Practice GMAT Problem Solving Question

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In a local intramural basketball league, there are 10 teams and each team plays every other team exactly one time. Assuming that each game is played by only two teams, how many games are played in total?

Correct Answer:

**D**- There are 10 teams, so any given team will play all of the other 9 teams, resulting in 9 games for each of the 10 teams.
- Multiply the 10 teams by the 9 games each team will play to yield 90 games. However, do not be tricked into picking answer B at this point: you are not yet done.
- 90 games includes duplicates (i.e., double-counting). In the counting of 90 games, we included the games team 1 played against teams 2 to 10. However, when we counted the number of games played by team 2, and likewise by teams 3 to 10, we also counted the games that these teams played against team 1.
- In other words, you cannot count as a unique game both Team 1 vs. Team 2 and Team 2 vs. Team 1. Based upon the question ("how many games are played in total"), we must find the number of unique games.
- Based upon this double-counting, 90 games is twice the number of games actually played. If we include only half the games we counted (including games such as team 1 vs team 2 while eliminating non-unique games such as team 2 vs team 1), we end up with a total count of 45 games (=90/2). The correct answer is D.

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