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 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
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For binary strings of length $n$, exactly half have even parity ($2^{n-1}$).
Updated On: Apr 10, 2026
- $2^{n-1}$
- $2^{n}-1$
- $2^{n-1}-1$
- $2^{n}$
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The Correct Option is A
Solution and Explanation
Step 1: Setup
The sequences with an even number of 0's can have 0, 2, 4, ... zeros.
Step 2: Binomial Sum
The total number of ways = $^nC_{0} + ^nC_{2} + ^nC_{4} + ...$
Step 3: Identity
It is a standard binomial identity that the sum of even-indexed coefficients equals $2^{n-1}$.
Final Answer: (a)
The sequences with an even number of 0's can have 0, 2, 4, ... zeros.
Step 2: Binomial Sum
The total number of ways = $^nC_{0} + ^nC_{2} + ^nC_{4} + ...$
Step 3: Identity
It is a standard binomial identity that the sum of even-indexed coefficients equals $2^{n-1}$.
Final Answer: (a)
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