Counting Question

A team has 22 players: 2 goalies, 10 only play defence and 10 only play forward. How many different teams of 11 players can be made if a team must have exactly one goalie, at least three forwards, and at least 3 on defence?

JellyFish said:

A team has 22 players: 2 goalies, 10 only play defence and 10 only play forward. How many different teams of 11 players can be made if a team must have exactly one goalie, at least three forwards, and at least 3 on defence?

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\(\displaystyle \sum\limits_{k = 0}^4 {2\left( {\begin{array}{c} {10} \\ {3 + k} \\ \end{array} } \right)\left( {\begin{array}{c} {10} \\ {7 - k} \\\end{array} } \right)}\)

Thank you very much,

I carried this out using each "pair", ie 10 choose 3 defenders with 10 choose 7 forwards, 10 choose 4 defenders and 10 choose 6 forwards etc until each pair was "flipped". I did all of the cominations, multiplied their sums by 2 and found that there are 361 008 possible teams of 11 players.

I just want to make sure: When I do 10 choose 5 defenders I also have 10 choose 5 forwards, I don't have to do this "pair" twice right?

If you use the sum I gave you above the 361008 is correct.
The 2 is from the number of ways to choose the goalie.