Combinatorics & the Pigeonhole Principle

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  • 0:05 Combinatorics
  • 1:30 The Pigeonhole Principle
  • 2:29 More Examples
  • 4:03 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll explore a branch of mathematics called combinatorics. We'll also look at a well-known principle in combinatorics called the pigeonhole principle and observe some simple examples in which this principle can be applied.


Suppose you decide to spend your Friday evening at a carnival. At the carnival, an announcer poses a problem and says that those who get the right answer will receive a small prize. Here's the problem the announcer poses: if you have 8 red gloves and 8 green gloves in a laundry basket for a total of 16 gloves, what's the minimum number of gloves you must pull out of the laundry basket to guarantee a matching pair?

You start to think about it, and know that it must be more than one glove since you need a matching pair. If you pulled out two gloves, you might get lucky and get a pair, but you could also get a red glove and a green glove, meaning you are not guaranteed a pair.

If you pull three gloves out of the laundry basket, each of the gloves could be red or green. If the first two gloves you pull are a match, you have your match. If not, you have a red and a green glove and since the third glove must be red or green, you would have to have a pair. So if you pull three gloves from the laundry basket, you're guaranteed a pair. You share your answer with the announcer: the minimum number of gloves that must be pulled from the laundry basket that would guarantee a matching pair is three, and collect your prize!

This glove problem is a math problem in the area of combinatorics. Combinatorics is a branch of mathematics that has to do with counting techniques. Don't let the big name scare you. As the definition states, it's just a fancy name for the study of counting.

The Pigeonhole Principle

The logic behind the glove problem actually has to do with an extremely simple but powerful concept in combinatorics. This concept is called the pigeonhole principle. Basically, this principle states that if we're placing pigeons in pigeonholes, and we have more pigeons than pigeonholes, it must be the case that at least one of the holes has more than one pigeon in it. Seems like common sense, right? It is, but you'd be surprised how useful this principle is in the study of combinatorics.

Think about our glove example again. Consider our gloves to be pigeons and the colors of gloves to be pigeonholes. If we pull three gloves out of the laundry basket, and those gloves can be one of two colors, red or green, we have three pigeons and two pigeonholes. By the pigeonhole principle, at least two of the pigeons must go in the same pigeonhole, so at least two of the gloves must be the same color when you pull out three. Pretty neat, isn't it? It's such a simple concept, but can be applied to so many different problems.

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