In how many ways can a two-person beach volleyball team be assembled such that no player feels out of position? 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?
This is my attempt:
If the one player wants to be a spiker we can choose any $1$ of $6$ people to be a spiker and any $1$ of $4$ people to be a setter, so $C(6,1) \cdot C(4,1) = 6 \cdot 4 = 24$ possible teams.  If the one player wants to be a setter, we can choose $1$ of $5$ people to be setters and $1$ of $5$ people to be spikers, so $C(5,1) \cdot C(5,1) = 5 \cdot 5 = 25$ possible teams. So, the total number of different teams $= 24 + 25 = 49$. But it's wrong...
Thanks in advance!
 A: Let's break down the cases in the way Math Lover suggested in the comments.
If the player who is willing to play either position is selected, then any of the other nine players may be selected to play the other position on the team.  Hence, $9$ such teams can be formed.
If the player who is willing to play either position is not selected, one of the five players who prefer to play spiker must be selected to play with one of the four players who prefer to be a setter, which can be done in $5 \cdot 4 = 20$ ways.
Since the two cases are mutually exclusive and exhaustive, there are
$$\binom{1}{1}\binom{9}{1} + \binom{5}{1}\binom{4}{1} = 9 + 24 = 29$$
ways to form a two-person beach volleyball team.
What was your mistake?
You counted each team in which the player who is willing to play either position does not play twice, once when you treated that player as a spiker and once when you treated that player as a setter.
To see this, suppose that if Angela is the player who is willing to play either position; Barbara, Claire, Daniela, Ella, and Fiona are spikers; and Gabriela, Helena, Ines, and Joanna are the setters.  For instance, among the $24$ teams you formed by treating Angela as a spiker is (Barbara, Gabriela) since Barbara is among the six potential spikers and Gabriela is among the four potential setters. However, the team (Barbara, Gabriela) is also among the $25$ teams you formed by treating Angela as a setter since Barbara is among the five potential spikers and Gabriela is among the five potential setters.
Since there are $\binom{5}{1}\binom{4}{1}$ teams on which Angela does not play, there are $$\binom{6}{1}\binom{4}{1} + \binom{5}{1}\binom{5}{1} - \binom{5}{1}\binom{4}{1} = 29$$ two-person beach volleyball teams which can be formed.
