Question

${\sum }_{r=0}^{n-1}\frac{{}^n{C}_r}{{}^n{C}_r{+}^n{C}_{r+1}}$=

• A. $\frac{n(n+1)}{2}$
• B. $\frac{n+1}{n-1}$
• C. $\frac{n+1}{2}$
• D. $\frac{n}{2}$

Answer 1

The correct option is B $\frac{n+1}{n-1}$

$\sum _{r=0}^{n=1}\frac{{}^n{C}_r}{{}^n{C}_r+{}^n{C}_r+1}=\sum _{r=0}^{n-1}\frac{{}^n{C}_r}{n+1}$

$\Rightarrow \frac{n!(n-r)!}{r!}\Rightarrow \sum _{r=0}^{n-1}\frac{(r+1)}{(n+1)}$

$\frac{(n+1)!(n-r)!}{(r+1)!}$

$=\frac{n+1}{n-1}$