SOLUTION: prove that: (n-r)(nCr)=n(n-1Cr)

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Question 21399: prove that: (n-r)(nCr)=n(n-1Cr)
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
(n-r)(nCr)=n(n-1Cr)
LHS=(n-r)(nCr)=(n-r)n!/{r!(n-r)!}=(n-r)n!/{r!(n-r)(n-r-1)!}=n!/{r!(n-r-1)!}
RHS=n(n-1Cr)=n(n-1)!/{r!(n-1-r)!}=n!/{r!(n-1-r)!}=lhs
note that we used n!=n(n-1)!here...since n!=n(n-1)(n-2)....3*2*1
=n(n-1)(n-2)....3*2*1=n[(n-1)(n-2)....3*2*1]=n(n-1)!