document.write( "Question 1107228: Proof that ncr=(n/n-r)(n-1 c r)\r
\n" ); document.write( "\n" ); document.write( "Proof that n(n-1 c r) = (r+1) (n c r+1) \n" ); document.write( "
Algebra.Com's Answer #722271 by math_helper(2461)\"\" \"About 
You can put this solution on YOUR website!
Part I
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\n" ); document.write( "\"+C%28n%2Cr%29+=+n%21%2F%28%28n-r%29%21%28r%21%29%29+\" (by definition)
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\n" ); document.write( " = \"+%28n%2A%28n-1%29%21%29%2F%28%28n-r%29%28n-r-1%29%21%2Ar%21%29%29+\"
\n" ); document.write( "We can bring the n into the factorial in the numerator to go from (n-1)! to n!,
\n" ); document.write( "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" ); document.write( " = \"++n%21%2F%28%28n-r%29%21%2Ar%21%29+\" DONE.
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\n" ); document.write( "Part II
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\n" ); document.write( "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+\"
\n" ); document.write( "RHS: (r+1)C(n,r+1) = \"+%28r%2B1%29%2An%21%2F%28%28n-r-1%29%21%28r%2B1%29%21%29+\"
\n" ); document.write( " = \"+%28r%2B1%29%2An%21%2F%28%28n-r-1%29%21%28r%2B1%29%28r%21%29%29+\"
\n" ); document.write( "Canceling (r+1) from numerator and denominator:
\n" ); document.write( "= \"+n%21%2F%28%28n-r-1%29%21%2Ar%21%29+\" = LHS, DONE
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