Online math tutoring platforms offer students new and exciting ways to learn their regular school topics. In class, lots of the concepts and principles that students have to learn can be tricky to grasp, often due to the way they are taught. Without adding a little bit of life and color to lessons, school teachers risk losing their students’ attention.With the right teaching methods, even elementary mathematical concepts can be interesting. One such concept is the Pigeonhole Principle, a simple but powerful concept that can be applied to many different mathematical fields. Students who are lucky enough to have a creative tutor outside of their school’s math class usually take to this topic like ducks to water.
What is the Pigeonhole Principle
Like other math principles and concepts that we’ve spoken about in recent months, the Pigeonhole Principle is a straightforward but deeply useful mathematical concept. The Pigeonhole Principle is an important principle in combinatorics, also known as the study of counting and arrangement, and understanding it works best with an example.
Imagine we have a number of pigeonholes and a group of pigeons. We want to distribute the pigeons amongst the pigeonholes. What the Pigeonhole Principle tells us is that if we have more pigeons than pigeonholes, at least one of our pigeonholes will contain more than one pigeon.
If we want to express this in mathematical terms, we can say that:
If n objects are distributed into m containers and n > m, then at least one container must contain more than one object.
As you can see, this principle is as useful as it is simple, and it is this simplicity that makes the Pigeonhole Principle an easy one to remember for young math students. Once kids grasp the concept, they will find out just how useful it can be, whether we want to use the principle for a wide variety of topics with varying degrees of difficulty, such as counting and probability.
Uses in Counting
On the most basic level, we can use the Pigeonhole Principle to help us with counting problems. By applying this principle to counting problems, we can make a seemingly difficult question very easy for ourselves.
Let’s imagine a group of 10 people who are about to begin shaking hands with one another. We want to find out how many handshakes will occur if every single person in the group shakes hands with every other person. Before frantically trying to figure out this problem, we can apply the Pigeonhole Principle.
If we want to do this, however, we must designate some pigeons and pigeonholes. We can label the people “pigeons” and the handshakes “pigeonholes”, and because everyone will shake hands with everyone else, we are distributing handshakes amongst individuals. We have 10 people, and every handshake involves 2 individuals.
Therefore, we can calculate the total number of handshakes by utilizing the formula:
Total handshakes = n⋅(n−1)/2
where n is the number of people. If we put our value for n=10 into this equation, we get:
Total handshakes = 10(10-1)/2 =45.
Now you can see how easy even some of the more complicated counting tasks can be when you properly understand and apply the Pigeonhole Theory.
Pigeonholes in Probability:
As laid out above, the applications of the Pigeonhole Principle stretch far beyond the realm of counting and into other topics, such as probability. We can use this same concept to identify the certainty of some given outcomes. I’m sure we would all agree that one of the most interesting concepts in probability is randomness, so let’s take a look at how the Pigeonhole Principle works in relation to randomness.
Let’s say you randomly distribute 56 bookmarks into 50 books. This time around, each bookmark represents a “pigeon”, while the books are the “pigeonholes”. The Pigeonhole Principle tells us that at a minimum, at least one book must contain more than one bookmark.
While this statement may seem obvious, it is important to understand what this means in probability terms. It is certain that at least one book will contain more than one bookmark, meaning the probability of at least one book containing more than one bookmark is 1, while the probability of each book containing a maximum of one bookmark is 0. These are definitive statements that we can make about chaotic events thanks to the Pigeonhole Principle.
At the Online Math Center, our tutors take pride in their years of experience and the new and fun methods they’ve learned to use to keep students engaged during their 1-1 tutoring lessons. We have the knowledge and expertise necessary to help math students boost their grades, whether they are struggling to pass or shooting for the stars.