Question

The arithmetic mean of the data \(0,1,2,\ldots,n\) with frequencies \(1,{}^nC_1,{}^nC_2,\ldots,{}^nC_n\) is

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

Hint:

Use binomial identities.

Answer 1

The Correct Option is D

Step 1: Mean: \[ \bar x=\frac{\sum x\cdot f}{\sum f} \]
Step 2: \[ \sum f=2^n,\quad \sum x f=n2^{n-1} \]
Step 3: \[ \bar x=\frac{n2^{n-1}}{2^n}=\frac n2 \]