The identity (n+1)Cr=nCr+nC(r-1), valid if n>=1 and 1=0. These facts embody the construction of Pascal's triangle and let you prove by induction (on n) that nCr is always an integer since it's either 1 or it's the sum of two integers.
You can also think of the combinatorial definition of nCr, the number of r-subsets of an n-set. This is definitely an integer. Of course then you'd have to prove that the factorial expression for nCr is correct...
The nCr are binomial coefficients with the property that n+1Cr = nCr-1 + nCr (think of Pascal's Triangle) with nCn = 1 = nC0. Since 0C0 = 1 it follows that all nCr are integers
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