Thread: combinatorial identity problem

1. combinatorial identity problem

Suppose that n people are gathered for a game requiring two teams each with k players. assume 2k <= n. 1 team wear red shirts and the other team will wear blue shirts. the n-2k people not on either team will just watch. we can choose the teams in either of 2 ways. 1 choose k players to wear the red shirts then choose k others to wear blue shirts. 2 the other is to choose the 2k players and then choose k of them to be red and the other will be blue. prove via combinatorial identity that these 2 methods are the same.

2. Originally Posted by canyiah
Suppose that n people are gathered for a game requiring two teams each with k players. assume 2k <= n. 1 team wear red shirts and the other team will wear blue shirts. the n-2k people not on either team will just watch. we can choose the teams in either of 2 ways. 1 choose k players to wear the red shirts then choose k others to wear blue shirts. 2 the other is to choose the 2k players and then choose k of them to be red and the other will be blue. prove via combinatorial identity that these 2 methods are the same.
The first method is $\binom{n}{k}\binom{n-k}{k}$.
The second method is $\binom{n}{2k}\binom{2k}{k}$.
Can you show they are the same?

3. i dont see it probably by the algebra method it will look the same.

4. Originally Posted by canyiah
i dont see it probably by the algebra method it will look the same.
I have no idea what that means.
Can you put into standard language?

5. ill try it later today thanks

6. showed they were equal via algebraic method just substitute using combinatorial formula c(n,k) = n!/(k!(n-k)!)