Question

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

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

Answer 1

The correct option is A $2^n$-1

The required no. of ways, i.e., the even no. of zeroes $\{0,2,4,........\}$

Therefore,

Required no. of ways $=\frac{n!}{n!}+\frac{n!}{2!(n-2)!}+\frac{n!}{4!(n-4)!}+.....={}^n{C}_0+{}^n{C}_2+{}^n{C}_4+.....=2^{n-1}$

Hence the correct answer is $\left(A\right)2^{n-1}$.