Question

The mean of the values $0,1,2,\dots \dots \dots ,n$ having corresponding weight ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2,\dots \dots \dots {\dots }^n{C}_n,$ respectively is

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

Answer 1

The correct option is D $\frac{n}{2}$

$\overline{x}=\frac{0.1+{1.}^n{C}_1+{2.}^n{C}_2.+{3.}^n{C}_3+n{.}^n{C}_n}{1{+}^n{C}_1{+}^n{C}_2+..........{+}^n{C}_n}$
$=\frac{\sum _{r=0}^{n}r{.}^n{C}_r}{\sum _{r=0}^n}=\frac{\sum _{r=1}^{n}r.\frac{n}{r}{}^{n-1}{C}_{r-1}}{\sum _{r=0}^n{}^n{C}_r}==\frac{n\sum _{r=1}^n{}^{n-1}{C}_{r-1}}{\sum _{r=0}^n{}^n{C}_r}$
$=\frac{n{.2}^{n-1}}{2n}=\frac{n}{2}$