Question

Your drawer has 5 pairs of black socks, 4 pairs of gray socks, 2 pairs of white socks, 1 pair of brown socks, and 1 pair of blue socks. the lights are out, and you're in a rush to leave. what is the smallest number of socks you should pull out so that you can be assured that you will have at least one pair of matching socks? note that there are no left or right socks and two socks of the same color will be considered matching.

Answer 1

5 pairs of black socks*2 because it’s a pair= 10 socks
4 pairs of grey socks*2=8
2 pairs of white socks*2=4
1 pair of brown socks*2=2
1 pair of blue socks*2=2
10 socks+8 socks+4 socks+2 socks+ 2 socks=26 socks in total
26/2=13+1 because you need to be sure you get one pair of socks. The smallest amount of socks you would have to grab to get at least one matching pair is 14 socks.

Answer 2

You can use pigeonhole principle to find the least number of socks to pull out to find at least one pair.

The smallest number of socks to pull out to be assured  that we get at least one pair of matching socks is 6.

What is pigeonhole principle?

If there are n objects to be putted into m containers such that n > m, then at least one container will have two objects.

How is pigeonhole principle going to save us in this situation?

Suppose the worst is happening when you're pulling the socks out.

The total number of colors is 5.

Now the worst case is in first five trials you didn't got any match (that means all 5 pulls were ending to be of different colors).

But now any more sock pulled will have to be falling into one of the five colors we already have. (this is exactly the pigeonhole principle's use)

Thus, total 6 pulls will ensure that at least one of the sock pair will be of same color.

Thus, one more thing you will notice, is that we don't need to know how many pairs of what color is there. Only the count of colors is enough.

Learn more about pigeonhole principle here:

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