Determining the Number of Ways to Pick Two Socks Without Forming a Pair

Imagine you're digging through a box containing 10 pairs of socks, each with a different color. Curiously, you wonder in how many different ways you can pick two socks one by one and ensure that they do not form a pair. This problem involves a combination of combinatorial mathematics and probability, fascinating for its simplicity and depth.

Total Ways to Pick Two Socks

First, let's determine the total number of ways to pick two socks from the box. Given that there are 20 socks in total, this can be calculated using the combination formula:

binom{20}{2} frac{20 times 19}{2 times 1} 190

This means there are 190 different ways to choose any two socks from the box.

Ways to Form a Pair

Next, we need to figure out how many of these 190 ways involve forming a pair. Since there are 10 pairs of socks, the number of ways to choose a pair from these 10 pairs is:

binom{10}{1} times binom{2}{1} 10 times 2 20

Thus, there are 20 ways to choose two socks that form a pair.

Ways Not to Form a Pair

To find the number of ways that the two socks do not form a pair, we subtract the number of ways to form a pair from the total number of ways to choose any two socks:

190 - 20 170

Therefore, there are 180 ways to pick two socks from the box such that they do not form a pair.

Alternative Methods

To verify this result, let's consider two alternative methods:

Method 1: Directly Counting Favorable Cases

In this method, we choose 2 pairs out of 10 pairs, and then choose one sock from each of these pairs. The calculation goes as follows:

binom{10}{2} times binom{2}{1} times binom{2}{1} 45 times 2 times 2 180

This confirms the previous result.

Method 2: Total Cases - Unfavorable Cases

In this method, we start with the total number of ways to pick any two socks and subtract the number of ways that they form a pair:

binom{20}{2} - binom{10}{1} 190 - 10 180

This is also consistent with our earlier findings.

Conclusion

In conclusion, there are 180 ways to pick two socks from a box containing 10 pairs of differently colored socks such that they do not form a pair. This problem is a delightful application of combinatorics and probability, showcasing the power of mathematical reasoning.