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

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) 2n-1 (B) 2n-1 (C) 2n-1-1 (D) 2n

Tardigrade Admin · Accepted Answer

The required number of ways = The even number of O's i.e., O, 2,4,6, ldots =(n !/n !)+(n !/2 !(n-2))+(n !/4 !(n-4) !) =n C0+ n C2+ n C4+ ldots=2n-1

Q. 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

$2^{n - 1}$

$2^{n} - 1$

$2^{n - 1} - 1$

$2^{n}$

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

Q. 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

2n−1