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 Two ways (Posted on 2013-10-31)
Show that if N is a positive integer, then C(2N,2)= 2*C(N,2)+ N^2

(a) using a combinatorial argument.
(b) by algebraic manipulation.

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 Ways Comment 2 of 2 |
a. Divide 2N things into 2 groups of N things. Choose 2 of them. If they are both from the first group, then there are C(N, 2) choices. If they are both from the second group, then there are C(N, 2) choices. If they are from different groups, then there are N^2 choices. In total, there are 2*C(N, 2)+N^2 choices. Therefore, C(2N, 2)=2*C(N, 2)+N^2.

b. C(N, 2)=N(N-1)/2
C(2N, 2)=2N(2N-1)/2=N(2N-1)=2N^2-N=(N^2-N)+N^2=N(N-1)+N^2=2*C(N, 2)+N^2

 Posted by Math Man on 2013-11-01 19:35:32

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