Difference between revisions of "2016 AMC 10B Problems/Problem 22"
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There are <math>21</math> teams. Any of the <math>\tbinom{21}3=1330</math> sets of three teams must either be a fork (in which one team beat both the others) or a cycle: | There are <math>21</math> teams. Any of the <math>\tbinom{21}3=1330</math> sets of three teams must either be a fork (in which one team beat both the others) or a cycle: | ||
Revision as of 14:38, 28 January 2020
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 1
There are teams. Any of the
sets of three teams must either be a fork (in which one team beat both the others) or a cycle:
But we know that every team beat exactly
other teams, so for each possible
at the head of a fork, there are always exactly
choices for
and
. Therefore there are
forks, and all the rest must be cycles.
Thus the answer is which is
.
Solution 2 (Cheap Solution)
Since there are teams and for each set of three teams there is a cycle, there are a total of
cycles of three teams. Because about
of the cycles
satisfy the conditions of the problems, our answer is close to
. Looking at the answer choices, we find that
is closer to
than any other answer choices, so our answer is
which is
.
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.