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Difference between revisions of "2016 AMC 10B Problems/Problem 22"

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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);
 
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>
 
</asy>
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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.
+
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> as A beat exactly 10 teams and we are choosing 2 of them. 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>\boxed{\textbf{(A)}}</math>.
 
Thus the answer is <math>1330-945=385</math> which is <math>\boxed{\textbf{(A)}}</math>.

Revision as of 14:45, 18 January 2021

Problem

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$

$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

Solution 1

There are $21$ teams. Any of the $\tbinom{21}3=1330$ 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 $10$ other teams, so for each possible $X$ at the head of a fork, there are always exactly $\tbinom{10}2$ choices for $Y$ and $Z$ as A beat exactly 10 teams and we are choosing 2 of them. Therefore there are $21\cdot\tbinom{10}2=945$ forks, and all the rest must be cycles.

Thus the answer is $1330-945=385$ which is $\boxed{\textbf{(A)}}$.

Solution 2 (Cheap Solution)

Since there are $21$ teams and for each set of three teams there is a cycle, there are a total of $\tbinom{21}3=1330$ cycles of three teams. Because about $1/4$ of the cycles $\{A, B, C\}$ satisfy the conditions of the problems, our answer is close to $1/4 \cdot 1330=332.5$. Looking at the answer choices, we find that $332.5$ is closer to $385$ than any other answer choices and that the next closest is$665$ which is twice $332.5$, so our answer is $385$ which is $\boxed{\textbf{(A)}}$.


Speaking of cheap solutions, you can let the teams be $1,2,3,..21$ and have $i$ beat $i+1,i+2, ... i+10 \pmod{21}$ and lose to $i+11,i+12, ... i+20 \pmod{21}.$ You can choose any random team as $A$ for $21$ ways but overcount since set, so divide by $3$ so multiply by $7.$ Now we can proceed w/casework (which isn't too hard). Eventually you realize the sum is $7(1+2+\cdots+10) = \boxed{\textbf{(A)}}$.

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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