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# counting pairs problem

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192
2
+62

It is a beautiful day at the beach and ten beach volleyball players have shown up at the volleyball courts. Each two-person volleyball team should consist of a setter and a spiker. Five of the players prefer to be a spiker, four of the players prefer to be a setter, and one player is fine either way.

In how many ways can a two-person team be assembled such that no player feels out of position?

Feb 24, 2021

#1
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There are 2 cases: one case where the player who is fine either way is not in the two-person team, or another case where that player is.

For case 1, there are $$5\cdot4=20$$ ways to select the teams, since there are 5 possibilities for a spiker and 4 for a setter.

For case 2, there are 9 ways to select the team, since the player who is fine either way can pair up with any of one of the 9 remaining people.

In total, there are $$20+9=\boxed{29}$$ ways.

Feb 24, 2021
#2
+62
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Thank you!

MysticBoba  Feb 24, 2021