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Ideal Gas Equation

n bit strings...

Question

$n$ bit strings are made by filling the digits $0$ or $1$ . The number of strings in which there are exactly $k$ zeros with not two $0$ 's consecutive is

^{(}^{n}^{-}^{k}^{)} C_{k}

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^{(}^{n}^{-}^{k}^{+}^{1}^{)} C_{k}

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^{(}^{n}^{-}^{k}^{-}^{1}^{)} C_{k}

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^{(}^{n}^{+}^{k}^{)} C_{k}

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Solution

The correct option is B $^{(}^{n}^{-}^{k}^{+}^{1}^{)} C_{k}$
Out of n bit strings, $k$ bits are zeros and $(n - k)$ bits are 1
First arrange (n - k) bits alternatively with 1 and fill (n - k + 1) places with k zeros
$∴$ Number of strings $=^{n}^{-}^{k}^{+}^{1} C_{k}$

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