The arithmetic mean of the following frequency distribution:
Variable (X): 0 1 2 3 ... n
Frequency (f): $^{n} C_{0}$ $^{n} C_{1}$ $^{n} C_{2}$ $^{n} C_{3}$ ... $^{n} C_{n}$
is

\frac{2^{n}}{n}

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\frac{2^{n}}{n + 1}

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\frac{n}{2}

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\frac{2}{n}

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Solution

The correct option is D $\frac{n}{2}$
Let $¯ ¯¯¯ ¯ X$ denote the required mean. Then,
$¯ ¯¯¯ ¯ X = \frac{n \sum r = 0 r .^{n} C_{r}}{n \sum r = 0^{n} C_{r}} = \frac{n \sum r = 1 r . \frac{n}{r}^{n - 1} C_{r - 1}}{n \sum r = 0^{n} C_{r}} = \frac{n n \sum r = 1^{n - 1} C_{r - 1}}{2^{n}}$
$\Rightarrow ¯ ¯¯¯ ¯ X = \frac{n \times 2^{n - 1}}{2^{n}} = \frac{n}{2} [∵ n \sum r = 1^{n - 1} C_{r - 1} = 2^{n - 1}]$

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Similar questions

\frac{^{n} C_{0}}{n} + \frac{^{n} C_{1}}{n + 1} + \frac{^{n} C_{2}}{n + 2} + \dots \dots + \frac{^{n} C_{n}}{2 n} =

$\frac{^{n} c_{0}}{n}$ + $\frac{^{n} c_{1}}{n + 1}$ + $\frac{^{n} c_{2}}{n + 2}$ +_ _ _ _ _ _ + $\frac{^{n} c_{n}}{2 n}$ =

T h e a r i t h m e t i c m e a n o f t h e s e r i e s^{n} c_{0},^{n} c_{1},^{n} c_{2} . . . . .,^{n} c_{n} i s :

{2.}^{n} C_{0} + 2^{2} . \frac{^{n} C_{1}}{2} + 2^{3} . \frac{^{n} C_{2}}{3} + \dots + 2^{n + 1} . \frac{^{n} C_{n}}{n + 1} =

The arithmetic mean of nC0,nC1,nC2....nCn is
(A)2^n/n
(B)2^n-1/n
(C)2^n/n+1
(D)2^n-1/n+1

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Variable (X):	0	1	2	3	...	n
Frequency (f):	$^{n} C_{0}$	$^{n} C_{1}$	$^{n} C_{2}$	$^{n} C_{3}$	...	$^{n} C_{n}$