A man is camping. He has packed 8 black socks, 8 blue socks, 8 white socks, and 8 gray socks. It is still dark when he gets dressed. How many socks would he have to pull out of his bag to BE SURE that he has a matching pair?
If he got a different color with each pull, the fifth sock will definitely match one of the first four.
To determine the number of socks the man would have to pull out of his bag to be sure he has a matching pair, we need to consider the worst-case scenario. Since it is still dark and he cannot see the colors of the socks, we assume that the first sock he pulls out can be of any color.
To guarantee a matching pair, the man would have to pull out a maximum number of socks for each different color before he gets a match. Since there are 8 socks of each color, the man would have to pull out 5 socks.
Here's the step-by-step breakdown:
1. The first sock can be any color.
2. The second sock he pulls out would guarantee a matching pair, as there are only 8 different colors at most.
3. However, to get the worst-case scenario, we assume he doesn't get a match yet, so he would have pulled out 1 sock.
4. With the third sock, he could either match the first or second sock he pulled out, no matter what colors they are. So, he either gets a match or pulls out a new color, totaling 2 socks pulled out.
5. Similarly, with the fourth sock, he matches one of the first three socks or pulls out a new color, totaling 3 socks pulled out.
6. The fifth sock guarantees a matching pair, even if all the previous socks were different colors.
Therefore, the man would have to pull out a minimum of 5 socks to be sure that he has a matching pair.
Please note that this worst-case scenario assumes there are no two socks of the same color initially, as all the colors are mentioned together.