A) \[n+1\]
B) \[\frac{n}{2}\]
C) \[n+2\]
D) None of these
Correct Answer: B
Solution :
\[\sum\limits_{r=0}^{n-1}{\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r}}+{{\,}^{n}}{{C}_{r+1}}}}=\sum\limits_{r=0}^{n-1}{\frac{1}{1+\,\frac{^{n}{{C}_{r+1}}}{^{n}{{C}_{r}}}}}=\sum\limits_{r=0}^{n-1}{\frac{1}{1+\frac{n-r}{r+1}}}\] \[=\sum\limits_{r=0}^{n-1}{\frac{r+1}{n+1}}=\frac{1}{n+1}\sum\limits_{r=0}^{n-1}{(r+1)}\]\[=\frac{1}{(n+1)}[1+2+...+n]=\frac{n}{2}\].
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