In a certain sock drawer, there are 4 pairs of black socks, 3 pairs of gray socks and 2 pairs of orange socks. If socks are removed at random without replacement, what is the minimum number of socks that must be removed in order to ensure that two socks of the same color have been removed? A. 4 B. 7 C. 9 D. 10 E. 11

To ensure that at least two socks of the same color have been removed, you can use the Pigeonhole Principle. The principle essentially states that if you are putting items into categories (pigeonholes), and there are more items than categories, at least one category must contain more than one item.

In this case, there are 3 categories (colors of socks): black, gray, and orange.

Now, let's calculate according to the worst-case scenario:

1. If you remove one sock, it could be of any color, say a black one. 2. When you remove the second sock, it could be a gray one. 3. The third sock you remove could be an orange one.

At this point, you have three socks, all of different colors, because there are three different colors available in the drawer.

4. Now, when you remove the fourth sock, no matter what color it is, it will match one of the previously removed socks (since there are only three colors to choose from).

So the minimum number of socks that must be removed to ensure that two socks of the same color have been removed is 4.

The correct answer is A. 4.

Extra: This problem demonstrates a principle in probability and combinatorics that is very handy in many real-life situations, not just theoretical math problems. The Pigeonhole Principle helps us predict outcomes where we're looking at grouping items together. It's useful in scenarios where we know the number of items and the number of groups or categories those items can belong to. Beyond mathematics, this principle can be applied in computer science (for instance, in hash tables), organizational tasks, resource allocation, and even in daily life situations such as reasoning why, if you have more people than bedrooms in a house, someone will have to share a room. Understanding the principle can be very helpful in thinking about problems logically and efficiently.