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) $\frac{n + 1}{2}$
B) $\frac{n - 1}{2}$
C) $\frac{2^{n} - 1}{2}$
D) $\frac{n}{2}$
Explanation
(d)
Mean=$\frac{0 \times^{n}C_{0} + 1 \times^{n}C_{1} + 2 \times^{n}C_{2} +_{\ldots} + n \times^{n}C_{n}}{\ ^{n}c_{0} +^{n}C_{1} +^{n}C_{2} +_{\ldots}\ +^{n}C_{n}}$
$= \frac{0 + 1{\times^{\ }}^{n}C_{1} + 2 \times^{n}C_{2} +_{\ldots} + n \times^{n}C_{n}}{2^{n}}$
$= \frac{n.2^{n - 1}}{2^{n}} = \frac{n}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.