# Combinatorial Proof Help?

Prove that:

C(n+k)C(n,k) = C(n+k,n-k)C(2k,k)

Update:

That should be for C(n+k,k)

### 2 Answers

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- kbLv 75 years agoFavorite Answer
Simply use the definition of the binomial coefficient, and rearrange factors as necessary.

C(n+k, k) C(n, k)

= ((n+k)!/(k! n!)) * (n!/(k! (n-k)!))

= (n+k)!/(k! k! (n-k)!)

= ((n+k)!/((n-k)! (2k)!)) * ((2k)!/(k! k!))

= C(n+k, n-k) C(2k, k).

I hope this helps!

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- TompLv 75 years ago
The first factor is missing a parameter.

- I am5 years agoReport
That should be for C(n+k,k)

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