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# Prove that C(n,k) = C(n,n-k)?

How do I prove that C(n,k) = C(n,n-k) using the theorem C(n,k) = n!/{k!(n-k)!} ?

And if you could explain with words also that would be helpful :) Thanks

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- Randy PLv 76 years agoFavorite Answer
Take the expression n-k and put in in place of k. C(a, b) means take the formula for C(n, k), use a in place of n and b in place of k.

So in place of k write (n-k). In place of n - k write n - (n-k).

C(n, n-k) = n! / [(n-k)! (n - (n-k))!]

Now work out what n - (n-k) is.

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- TomVLv 76 years ago
C(n,k) = n!/[k!(n-k)!]

C(n,n-k) = n!/[(n-k)!(n-n+k)!] = n![(n-k!)k!] = C(n,k)

If two things are equal to the same thing, then they are equal to each other.

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