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Question

Mathematics Question on Binary operations

A binary sequence is an array of 0′s0's and 1′s1's. The number of nn -digit binary sequences which contain even number of 0′s0's is

A

2n−12^{n-1}

B

2n−12^{n}-1

C

2n−1−12^{n-1}-1

D

2n2^{n}

Answer

2n−12^{n-1}

Explanation

Solution

The required number of ways == The even number of 0's
ie, 0,2,4,6,…\\{0,2,4,6, \ldots\\}
=n!n!+n!2!(n−2)!+n!4!(n−4)!=\frac{n !}{n !}+\frac{n !}{2 !(n-2) !}+\frac{n !}{4 !(n-4) !}
=nC0+nC2+nC4+…=2n−1={ }^{n} C_{0}+{ }^{n} C_{2}+{ }^{n} C_{4}+\ldots=2^{n-1}