Q. The mean of the values $0,\ 1,\ 2,\ 3,\ ...\ n$ with the corresponding weights ${\ ^{n}C}_{0},{\ ^{n}C}_{1},\ldots,{\ ^{n}C}_{n}$ respectively is
A) $\frac{2^{n}}{(n + 1)}$
B) $\frac{2^{n + 1}}{n(n + 1)}$
C) $\frac{n + 1}{2}$
D) $\frac{n}{2}$
Explanation
(d)
Required mean $= \frac{0.{\ ^{n}C}_{0} + 1.{\ ^{n}C}_{1} + 2.{\ ^{n}C}_{2}... + n.{\ ^{n}C}_{n}}{{\ ^{n}C}_{0} + {\ ^{n}C}_{1} +^{n}C_{2} + ... + {\ ^{n}C}_{n}}$
$= \frac{n.2^{n - 1}}{2^{n}} = \frac{n}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.