SOLUTION: Proof that ncr=(n/n-r)(n-1 c r) Proof that n(n-1 c r) = (r+1) (n c r+1)

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Question 1107228: Proof that ncr=(n/n-r)(n-1 c r)
Proof that n(n-1 c r) = (r+1) (n c r+1)
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Part I
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+C%28n%2Cr%29+=+n%21%2F%28%28n-r%29%21%28r%21%29%29+ (by definition)

= +%28n%2A%28n-1%29%21%29%2F%28%28n-r%29%28n-r-1%29%21%2Ar%21%29%29+
We can bring the n into the factorial in the numerator to go from (n-1)! to n!,
and similarly we can bring in the cross%28n-1%29 (EDIT: n-r) in the denominator to go from (n-1-r)! to (n-r)! :
= ++n%21%2F%28%28n-r%29%21%2Ar%21%29+ DONE.
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Part II
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LHS: nC(n-1,r) = +n%2A%28n-1%29%21%2F%28%28n-1-r%29%21%2Ar%21%29+=+n%21%2F%28%28n-1-r%29%21%2Ar%21%29+
RHS: (r+1)C(n,r+1) = +%28r%2B1%29%2An%21%2F%28%28n-r-1%29%21%28r%2B1%29%21%29+
= +%28r%2B1%29%2An%21%2F%28%28n-r-1%29%21%28r%2B1%29%28r%21%29%29+
Canceling (r+1) from numerator and denominator:
= +n%21%2F%28%28n-r-1%29%21%2Ar%21%29+ = LHS, DONE