Difference between revisions of "2016 AMC 10B Problems/Problem 22"
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\textbf{(D)}\ 1140 \qquad | \textbf{(D)}\ 1140 \qquad | ||
\textbf{(E)}\ 1330</math> | \textbf{(E)}\ 1330</math> | ||
+ | |||
+ | ==Solution== | ||
+ | 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: | ||
+ | |||
+ | <asy>size(7cm);label("X",(5,5));label("Z",(10,0));label("Y",(0,0));draw((4,4)--(1,1),EndArrow);draw((6,4)--(9,1),EndArrow); | ||
+ | label("X",(20,5));label("Z",(25,0));label("Y",(15,0));draw((19,4)--(16,1),EndArrow);draw((16,0)--(24,0),EndArrow);draw((24,1)--(21,4),EndArrow); | ||
+ | </asy> | ||
+ | But we know that every team beat exactly <math>10</math> other teams, so for each possible <math>X</math> at the head of a fork, there are always exactly <math>\tbinom{10}2</math> choices for <math>Y</math> and <math>Z</math>. Therefore there are <math>21\cdot\tbinom{10}2=945</math> forks, and all the rest must be cycles. | ||
+ | |||
+ | Thus the answer is <math>1330-945=385</math> which is <math>\textbf{(A)}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2016|ab=B|num-b=21|num-a=23}} | ||
+ | {{MAA Notice}} |
Revision as of 12:08, 21 February 2016
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
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 .
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 |
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