2013 AMC 10A Problems/Problem 24

Revision as of 14:13, 20 January 2015 by Saurabh g (talk | contribs) (Solution)

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

$\textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900$


  • Credit saurabh_g for this solution

Let us label the players of the first team $A$, $B$, and $C$, and those of the second team, $X$, $Y$, and $Z$. One way of scheduling all six rounds could be:

Round1---> AX BY CZ
Round2---->AX BZ CY
Round3---->AY BX CZ
Round4---->AY BZ CX
Round5---->AZ BX CY
Round6---->AZ BY CX

The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule by permuting those 6 rounds and that can be done in $6!$=$720$ ways.

So the answer is C) $720$

See Also

2013 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AMC 10 Problems and Solutions

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