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Question

In the usual notations prove that
(C0+C1)(C1+C2).....(Cn1+Cn)
=(n+1)nn!C1C2C3.....Cn

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Solution

We have prove that
(C0+C1)C1(C1+C2)C2......Cn1+CnCn=(n+1)nn!
or (C0C1+1)(C1C2+1).....(Cn1Cn+1)=(n+1)nn!
Now nCrnCr1=nr+1r=(n+1)r1
CrCr1+1=(n+1)r or Cr+Cr1Cr1=(n+1)r
Putting r = 1, 2, 3 ,........n in both sides and multiply etc
Cn=C0

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