Calculating Combinations of Socks Selection with Replacement

Introduction

Imagine a scenario where a bag contains a total of 6 pairs of socks. The objective is to determine the number of possible outcomes when selecting 2 pairs of socks from the bag with replacement. This article will delve into the mathematical concepts and steps required to solve this problem. We will explore the principles of combinations and probability to arrive at the solution. Understanding these concepts is essential for various applications in mathematics, including machine learning and data analysis.

Step 1: Understanding the Problem

The problem involves selecting 2 pairs of socks from a set of 6 pairs. Importantly, the socks are selected with replacement, meaning that after each selection, the socks are placed back into the bag before the next selection. This ensures that the total number of socks remains constant for each selection.

Step 2: Calculating the Number of Choices for Each Selection

In this scenario, for each individual sock pair, there are 6 possible choices. This is because there are 6 pairs of socks in total.

Selection of the First Pair

For the first selection, you have 6 choices. This is straightforward since there are 6 pairs of socks:

[ text{Choices for 1st Pair} 6 ]

Selection of the Second Pair

After the first pair has been selected, it is placed back into the bag. Hence, the total number of choices remains 6 for the second selection:

[ text{Choices for 2nd Pair} 6 ]

Step 3: Calculating the Total Number of Outcomes

To find the total number of possible outcomes, we need to multiply the number of choices for each individual selection. Since the selections are independent of each other due to replacement, the total number of outcomes is:

[ text{Total Outcomes} text{Choices for 1st Pair} times text{Choices for 2nd Pair} 6 times 6 36 ]

Therefore, the total number of outcomes when selecting 2 pairs of socks with replacement is 36.

Step 4: Understanding the Concept of Combinations

In the provided solution, the concept of combinations is used to arrive at the same result. Combinations are used when the order of selection does not matter. The notation nCr represents the number of ways to choose r items from a set of n items without regard to order. In our case, selecting 1 pair from 6 pairs can be represented as 6C1.

Since we are making two selections, and each selection has 6 choices, the total number of outcomes using combinations can be calculated as:

[ 6C1 times 6C1 6 times 6 36 ]

Conclusion

Through rigorous mathematical analysis, we have determined that the number of possible outcomes when selecting 2 pairs of socks from a bag containing 6 pairs with replacement is 36. This problem serves as a fundamental example in understanding the principles of combinations and probability theory. These concepts are crucial in various fields, including statistics, machine learning, and data analysis.