Q43 / 120 · VITEEE 2009 · Mathematics

A binary sequence is an array of 0 's and l's. The number of $n$-digit binary sequences which contain even number of 0 's is

Correct answer
A
$2^{n-1}$
B
$2^{n}-1$
C
$2^{n-1}-1$
D
$2^{n}$

Solution

The required number of ways $=$ The even number of $0^{\prime}$ s i.e., $\{0,2,4,6, \ldots\}$ $$ \begin{array}{l} =\frac{n !}{n !}+\frac{n !}{2 !(n-2) !}+\frac{n !}{4 !(n-4) !} \\ ={ }^{n} C_{0}+{ }^{n} C_{2}+{ }^{n} C_{4}+\ldots=2^{n-1} \end{array} $$

Hence, the correct answer is option: 'A'.

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