Question

A string consisting of \(0^{\prime}\) s and 1 's has even parity if 1 occurs an even number of times: otherwise, the string has odd parity. How many strings of length \(n\) have even parity? How many strings of length \(n\) have odd parity?

Short answer

\(2^{n-1}\) strings have even parity and \(2^{n-1}\) have odd parity.

Step-by-step solution

Understanding the Problem

A string has even parity if the number of 1's in the string is even, and odd parity if the number of 1's is odd. We are required to find how many strings of length \( n \) have even and odd parity.

Counting All Possible Strings

For any string of length \( n \), each position in the string can either be a 0 or a 1. Thus, there are a total of \( 2^n \) possible strings of length \( n \).

Counting Even Parity Strings

Strings with even parity mean there is an even number of 1's in the string. When counting strings of even parity, we notice that for each string with even parity, there is a corresponding string with odd parity by simply flipping one bit. Therefore, the number of strings with even parity is exactly half of the total number of strings, which is \( \frac{2^n}{2} = 2^{n-1} \).

Counting Odd Parity Strings

Similarly, strings with odd parity make up the other half of all possible strings. Thus, the number of strings with odd parity is also \( \frac{2^n}{2} = 2^{n-1} \).

Conclusion

For any string of length \( n \), there are \( 2^{n-1} \) strings with even parity and \( 2^{n-1} \) strings with odd parity, together making up all possible strings.

Key concepts

Binary Strings

In computer science and digital logic, binary strings refer to sequences composed solely of the characters '0' and '1'. These binary strings are the building blocks of binary code, which is extensively used in computer systems for data representation and processing.
Every character in a binary string can either be a '0' or a '1'. When considering a binary string of length \( n \), there are \( 2^n \) possible combinations of '0's and '1's.
This is because there are two choices for each position in the string, and these choices can vary independently from other positions.
Binary strings not only play a vital role in computing but are also crucial in mathematical problems related to combinatorics and coding theory.

Parity

The concept of parity is essential in understanding how data can be organized and interpreted within binary strings. Parity refers to whether the number of '1's in a binary string is odd or even.
A string is said to have even parity if it contains an even number of '1's. Conversely, it has odd parity if the count of '1's is odd.

• An example of a string with even parity is "1100" (containing two '1's).
• An example of a string with odd parity is "101" (containing two '1's).

Understanding parity is essential for error detection in computing systems and communication protocols.
By simply examining the number of '1's, parity can help ensure data was transmitted correctly or spot errors in binary representations.

Counting Techniques

When calculating the number of binary strings of a given parity, specific counting techniques are applied. Given a string of length \( n \), calculating how many of these configurations contain even or odd numbers of '1's boils down to dividing the total number of combinations.
We already know there are \( 2^n \) total possible binary strings for a given length \( n \). To find how many have even parity, one realizes that for every string, inverting its bits toggles its parity – meaning turning a '1' into a '0', or a '0' into a '1' alters whether its parity is odd or even.
Therefore, there is an equal number of even and odd parity strings among all \( 2^n \) possibilities. Thus, both categories hold \( \frac{2^n}{2} = 2^{n-1} \) strings.

• For strings of even parity, where the number of '1's is even, there are \( 2^{n-1} \) strings.
• For strings of odd parity, the total is similarly \( 2^{n-1} \).

Applying these simple counting techniques allows for a structured approach to determine the organization of binary sequences that meet specific parity conditions.
This knowledge is useful in coding theory and data encryption, ensuring data integrity and efficiency.