∑ _ n=1 ^ r=1 n C_ r [ n C_ r +n C_ (r+1) ] #160; #160; ⇒ ^ 1 C_ o ^ 1 C_ o +^ 1 C_ 1 = 1+1 1+1 = 1 2 #160; ...

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Combination

∑n-1r=0nCrnCr...

Question

$\sum_{n - 1}^{r = 0} \frac{^{n} C_{r}}{^{n C_{r} +^{n C_{r} + 1}}} =$

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Solution

$\begin{matrix} \sum_{n = 1}^{r = 1} \frac{n C_{r}}{[n C_{r} + n C_{(r + 1)}]} \Rightarrow \frac{^{1} C_{o}}{^{1} C_{o} +^{1} C_{1}} = \frac{1 + 1}{1 + 1} = \frac{1}{2} [∵^{n} C_{o} =^{n} C_{n} = 1] \end{matrix}$

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Similar questions

n \sum r = 0 {(\frac{^{n} C_{r - 1}}{^{n} C_{r} +^{n} C_{r - 1}})}^{3} = \frac{25}{24},

n

$^{n} C_{r} +^{n} C_{r - 1}$ is equal to

^{n} C_{r} + {2.}^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}

If n is an even natural number , then $n \sum r = 0 \frac{(- 1)^{r}}{^{n} C_{r}} e q u a l s :$

^{n} C_{r - 1} = 36,^{n} C_{r} = 84,^{n} C_{r + 1} = 126

(n, r)

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