Paul has 8 white socks, 4 blue socks,10 grey socks, and 12 black socks in his drawer. The socks are all jumbled up. What is the greatest number of socks that he would need to pull out to have a matching pair?
The greatest number of socks to get a matching pair is 5. If your first pull is white, your second blue, your third grey, and your fourth black, then the fifth sock you pull has to be one of those same colors, making a pair.
To determine the greatest number of socks that Paul would need to pull out to have a matching pair, we can use the concept of the Pigeonhole Principle.
The Pigeonhole Principle states that if there are n+1 objects placed into n containers, then there must be at least one container with two or more objects.
In this case, Paul has 8 white socks, 4 blue socks, 10 grey socks, and 12 black socks. To find the greatest number of socks Paul would need to pull out to have a matching pair, we want to consider the worst-case scenario.
The worst-case scenario is when Paul first selects one sock of each color. This means he has pulled out 4 socks (1 white, 1 blue, 1 grey, and 1 black) without having a matching pair.
Next, to ensure that he gets a matching pair after pulling out more socks, Paul would need to pull out socks of the same color as the ones already chosen. Since he has already chosen 1 sock of each color, he would need to pull out at least one more sock from each color to guarantee a matching pair.
Therefore, the greatest number of socks that Paul would need to pull out to have a matching pair is:
1 white (already chosen) + 1 white (to ensure a matching pair) +
1 blue (already chosen) + 1 blue (to ensure a matching pair) +
1 grey (already chosen) + 1 grey (to ensure a matching pair) +
1 black (already chosen) + 1 black (to ensure a matching pair) = 8 socks.
So, the greatest number of socks that Paul would need to pull out is 8 socks.