SOLUTION: Prove that nCr=nC(n-r).

Algebra ->  Permutations -> SOLUTION: Prove that nCr=nC(n-r).      Log On


   

Question 1019557: Prove that nCr=nC(n-r).
Found 2 solutions by Alan3354, mathmate:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Prove that ncr = ncn-r .
===========
In general, they're not equal.

Answer by mathmate(429) About Me  (Show Source):
You can put this solution on YOUR website!

Question:
Prove that nCr=nCn-r.

Solution:
By definition,
nCr=n%21%2F%28r%21%2A%28n-r%29%21%29
Substitute n-r for r, then
nCn-r
=n%21%2F%28%28n-r%29%21%2A%28n-%28n-r%29%29%21%29
=n%21%2F%28%28n-r%29%21%2Ar%21%29
=n%21%2F%28r%21%2A%28n-r%29%21%29 [by commutativity of multiplication]

Since the simplified expressions of
nCr and nCn-r are equivalent, therefore nCr=nCn-r.