Question

The value of $\sum _{r=0}^4\frac{{}^4{C}_r}{{}^4{C}_r+{}^4{C}_{r+1}}$ is

• A. $\frac{1}{2}$
• B. $3$
• C. $\frac{1}{3}$
• D. $2$

Answer 1

The correct option is B $3$

$\sum _{r=0}^4\frac{{}^4{C}_r}{{}^4{C}_r+{}^4{C}_{r+1}}=\sum _{r=0}^4\frac{{}^4{C}_r}{{}^5{C}_{r+1}}=\sum _{r=0}^4\frac{r+1}{5}(\because {}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r)=\frac{1}{5}\sum _{r=0}^{4}r+1=\frac{1}{5}(1+2+3+4+5)=3$