Introduction: In the world of digital information, strings of bits play a fundamental role. This article delves into the combinatorial problem of finding how many 4-bit binary strings contain exactly two 1s and two 0s. We explore this problem from a mathematical perspective, using both combinatorial methods and direct enumeration.
Understanding Bit Strings
A bit string is a sequence of bits (binary digits), which are either 0 or 1. These strings are crucial in computer science, digital communications, and data storage systems. When we talk about 4-bit strings, we are considering sequences of 4 positions, each of which can hold either a 0 or a 1.
The Problem: 4-bit Strings with 2 Ones and 2 Zeros
Our specific problem is to determine the number of 4-bit strings that contain exactly two 1s and two 0s. This is a classic example of a combinatorial problem, where we are interested in the number of ways to arrange specific elements within a given set.
Combination Approach
To solve this problem using combinatorial methods, we need to understand the concept of combinations. A combination is a selection of items from a larger set, such that the order of selection does not matter. In our case, we are selecting 2 positions out of 4 to place the 1s, and the remaining 2 positions will automatically be filled with 0s.
The number of ways to choose 2 positions out of 4 is given by the binomial coefficient, often denoted as "4 choose 2." Mathematically, this is expressed as:
C(4, 2) 4! / (2! * (4-2)!) 6
Let’s break this down step by step:
4! (4 factorial) is the product of all positive integers up to 4, which is 4 * 3 * 2 * 1 24. 2! is 2 * 1 2. (4-2)! 2!, which is again 2 * 1 2.Substituting these values into the formula:
C(4, 2) 24 / (2 * 2) 24 / 4 6
Therefore, there are 6 possible ways to arrange 2 ones and 2 zeros in a 4-bit string.
Permutation and Direct Enumeration
In addition to using combinations, we can also solve this problem by direct enumeration. This involves listing all possible 4-bit strings that contain exactly two 1s and two 0s. Here are the six possible arrangements:
0011 0101 0110 1001 1010 1100As we can see, each string has exactly two 1s and two 0s, and there are no other possibilities. This confirms our earlier combinatorial result.
Summary and Applications
In conclusion, the number of 4-bit strings that contain exactly two 1s and two 0s is 6. This result is obtained using both combinatorial methods (combinations) and direct enumeration. Understanding these concepts is crucial in various areas of computer science and information theory, such as error detection and correction, data compression, and digital signal processing.
Whether you are designing a digital communication protocol or developing a new algorithm for data storage, the ability to count and manipulate bit strings is essential. By mastering these fundamental concepts, you can tackle more complex problems with confidence and precision.