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I could use some assistance with understanding this problem.

Proof that I need assistance with

I understand that there are ${n}\choose{k}$ is a representation of ${n}\choose{k}$ ways to choose k elements from a set of n elements, but I don't understand how this can be represented as a integer. Could anyone point me in the right direction, because I'm not understanding this proof. Thank you!

Earthbound27
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    There are a few proofs I have seen of this, my favorite being using p-adic analysis. But I think if this is a logic class, the fact that it used to "count" things is probably sufficient. – operatorerror Apr 28 '16 at 03:26
  • Are you allowed to just manipulate factorials? – Xoque55 Apr 28 '16 at 03:28
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    If the $\binom{n}{k}$ in the text used is defined as a number of subsets it is clearly an integer. But maybe the text defines it using the factorial definition, and then some proof is needed. – coffeemath Apr 28 '16 at 03:30
  • for the binomial theorem, it is obvious that $(a+b)^n = \sum_{k=0}^n c_{n,k} a^k b^{n-k}$ where $c_{n,k}$ are integers, what needs a proof is that $c_{n,k} = \frac{n!}{k! (n-k)!}$ – reuns Apr 28 '16 at 03:42
  • @Xorque55 Yes I believe so. – Earthbound27 Apr 28 '16 at 04:23
  • @coffeemath I believe it is a factorial proof, it is not clear in terms of methods. – Earthbound27 Apr 28 '16 at 04:23
  • @user1952009 how would you prove that? I don't totally understand what you did. – Earthbound27 Apr 28 '16 at 04:24
  • how would I prove that $(a+b)^2 = a^2 + 2ab + b^2$ ? and hence that $(a+b)^3 = (a^2 + 2ab + b^2)(a+b) = a^3+3a^2b+3ab^2 + b^3$ ? and hence that $(a+b)^4 = (a^3+3a^2b+3ab^2 + b^3)(a+b) = \ldots$ ? – reuns Apr 28 '16 at 04:26
  • I believe there is an answer here: http://math.stackexchange.com/questions/11601/proof-that-a-combination-is-an-integer – yoyostein Apr 28 '16 at 04:34

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