If C0-C1C1-C2 -Cn-1-Cn-k C0 C1 C2- Cn then k-

The correct option is C (n+1)^n/n!k=(C_0+C_1)(C_1+C_2)…(C_n+1+C_n)/C_0C_1… C_n=(n 1)C_1(n+1)C_2…(n+1)C_n/^nC_0^nC_1…^nC_n=(n+1)^n/n! ...

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Proof by mathematical induction

If C0+C1C1+C2...

Question

If $(C_{0} + C_{1}) (C_{1} + C_{2}) \dots (C_{n - 1} + C_{n}) = k C_{0} C_{1} C_{2} \dots C_{n}$ then k =

\frac{(n + 1)^{n}}{n!}

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\frac{(n + 1)}{n!}

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\frac{(n + 2)^{n}}{n!}

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\frac{(n + 1)}{n!}

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Solution

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C_{r} =^{n} C_{r}

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(C_{0} + C_{1}) (C_{1} + C_{2}) (C_{2} + C_{3}) (C_{3} + C_{4}) . . . . . (C_{n - 1} + C_{n}) = \frac{C_{0} C_{1} C_{2} . . . . C_{n - 1} {(n + 1)}^{n}}{n!}

\frac{(C_{0} + C_{1}) (C_{1} + C_{2}) (C_{2} + C_{3}) . . . . . (C_{n - 1} + C_{n})}{C_{0} + C_{1} (C_{2} . . . . . . C_{n})} =

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