Question

The mean of the values $0,1,2,.....,n$ with the corresponding weights ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2,...{.}^n{C}_n$ respectively is

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

Answer 1

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

${(1+x)}^n{=}_0^{n}C{+}_1^{n}C.x{+}_2^{n}C.x^2+...{+}_n^{n}C.x^n$
Differentiating both sides:
$n{(1+x)}^{n-1}=0{+}_1^{n}C{+}_2^{n}C.2x+...{+}_n^{n}C.n{x}^{n-1}$
Substituting $x=1$:
$n{.2}^{n-1}=0{+}_1^{n}C{+}_2^{n}C.2+...{+}_n^{n}C.n$
Also, $2^n{=}_0^{n}C{+}_1^{n}C{+}_2^{n}C+...{+}_n^{n}C$
Thus, the required value $=\frac{{}_0^{n}C.0{+}_1^{n}C.1{+}_2^{n}C.2+...{+}_n^{n}C.n}{{}_0^{n}C{+}_1^{n}C{+}_2^{n}C+...{+}_n^{n}C}=\frac{n{.2}^{n-1}}{2^n}=\frac{n}{2}$
Hence, (D) is correct.