The weighted mean of first n natural numbers whose weights are equal to the number of selections out of n natural numbers of corresponding numbers, is

A
$\frac{n \cdot 2^{n - 1}}{2^{n} - 1}$
B
$\frac{3 n (n + 1)}{2 (2 n + 1)}$
C
$\frac{(n + 1) (2 n + 1)}{6}$
D
$\frac{n (n + 1)}{2}$

Solution:

The required mean

$\begin{matrix} \bar{x} = \frac{1 \cdot^{n} C_{1} + 2 \cdot^{n} C_{2} + 3 \cdot^{n} C_{3} + \dots + n \cdot^{n} C_{n}}{^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n}} \\ = \frac{\sum_{r = 0}^{n} r \cdot^{n} C_{r}}{\sum_{r = 1}^{n}^{n} C_{r}} = \frac{\sum_{r = 1}^{n} r \cdot {\frac{n}{r}}^{n - 1} C_{r - 1}}{\sum_{r = 1}^{n}^{n} C_{r}} \\ = \frac{n \cdot 2^{n - 1}}{2^{n} - 1} \end{matrix}$

Solution:

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