2016 AMC 12B Problems/Problem 20
Problem
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost
games; there were no ties. How many sets of three teams
were there in which
beat
,
beat
, and
beat
Solution
We use complementary counting. Firstly, because each team played other teams, there are
teams total. All sets that do not have
beat
,
beat
, and
beat
have one team that beats both the other teams. Thus we must count the number of sets of three teams such that one team beats the two other teams and subtract that number from the total number of ways to choose three teams.
There are ways to choose the team that beat the two other teams, and
to choose two teams that the first team both beat. This is
sets. There are
sets of three teams total. Subtracting, we obtain
,
as our answer.
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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All AMC 12 Problems and Solutions |
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