Question

A binary sequence is an array of $0's$ and $1's$. The number of $n$-digit binary sequences which contain even number of $0's$ is

• A. $2^{n-1}$
• B. $2^{n}-1$
• C. $2^{n-1}-1$
• D. $2^{n}$

Answer 1

Answer: (A)

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