Prove that C0 - C1C1 - C2C2 - C3C3 - C4-Cn - 1 - Cn - C0C1C2-Cn - 1n - 1nn-

LHS= (C_0+C_1)(C_1+C_2)(C_2+C_3)....(C_n 1+C_n) =∏_r=1^n(C_r 1+C_r) =∏_r=1^n ^n+1C_r(∵ ^nC_r+^nC_r 1=^n+1C_r) =∏_r=1^n(n+1r)^nC_r 1 ...

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Greatest Binomial Coefficients

Prove that C...

Question

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

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c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

{(1 + x)}^{n}

(c_{0} + c_{1}) (c_{1} + c_{2}) . . . . . . . (c_{n - 1} + c_{n}) = \frac{c_{1} c_{2} . . . c_{n} {(n + 1)}^{n}}{| n - -}

(C_{0} + C_{1}) (C_{1} + C_{2}) . . . . . (C_{n - 1} + C_{n})

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

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

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n}

(C_{0} + C_{1}) (C_{1} + C_{2}) \dots (C_{n - 1} + C_{n}) = k \cdot C_{1} C_{2} C_{3} \dots C_{n}

k

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