A binary sequence is an array of 0's and 1's. | vedclass

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(A) Let $S$ be the set of all $n$-digit binary sequences. The total number of such sequences is $2^n$. Let $E$ be the number of sequences with an even

Vedclass · Accepted Answer

$2^{n-1}$

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(A) Let $S$ be the set of all $n$-digit binary sequences. The total number of such sequences is $2^n$. Let $E$ be the number of sequences with an even number of $0$'s and $O$ be the number of sequences with an odd number of $0$'s. We know that the sum of binomial coefficients $\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + . . . + \binom{n}{n} = 2^n$. The number of sequences with an even number of $0$'s is given by the sum of even-indexed binomial coefficients: $E = \binom{n}{0} + \binom{n}{2} + \binom{n}{4} + . . .$. Using the property of binomial coefficients,$\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + . . . = 2^{n-1}$. Thus,the number of $n$-digit binary sequences containing an even number of $0$'s is $2^{n-1}$.

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

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