The arithmetic mean of $^{n} C_{0},^{n} C_{1},^{n} C_{2}, . . .,^{n} C_{n}$ is

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

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

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

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

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Solution

The correct option is D $\frac{2^{n}}{n + 1}$
Required mean $= \frac{^{n} C_{0} +^{n} C_{1} + . . . . . . . . . . . . . .^{n} C_{n}}{n + 1} = \frac{(1 + 1)^{n}}{n + 1} = \frac{2^{n}}{n + 1}$

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Similar questions

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 :

\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} =

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

{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} =

0, 1, 2, 3, \dots, n

^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}

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