Combinatorics: Definition & History

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson looks at the branch of mathematics known as combinatorics. We will look at the definition of combinatorics as well as some of its history and how it is used today. When you are finished, you can test your new found knowledge with a quiz.

Combinatorics

Combinatorics is a branch of mathematics that may sound a bit intimidating, but in fact, is just a fancy name for counting techniques. Combinatorics can be used to combine objects using rules to create new arrangements of those objects, count the number of arrangements that can be made from a group of objects, and find the best arrangement of objects given different circumstances.

Two combinatorics categories are enumeration, which involves counting methods, and graph theory, which involves the study of graphs. Both of these categories use permutations and combinations often. A permutation is an arrangement of objects where order matters. For example, BAC is a permutation of the letters ABC. It is basically an ordering of a group of objects.

A combination is a group of objects, where order does not matter. For example, suppose there are twenty students in a class, and you choose five of them to go to a math challenge. Regardless of the order you name these students, they still represent the same group. Those 5 students represent a combination of five of the twenty students in the class.

You may have heard of permutations and combinations in the study of probability. They are intricately involved with the study of probability. Probability looks at how likely certain outcomes are to happen, and it involves a lot of calculating and counting how many ways a certain event can happen. With counting involved, combinatorics is involved, and we see that permutations and combinations are an important part of combinatorics.

History of Combinatorics

Combinatorics has been around for quite some time. We can date the concept of combinatorics back as far as 2800 B.C.E. Chinese literature introduces the earliest known magic square around this time. A magic square is an array of numbers in a square such that the sum of the rows, columns, and diagonals are all the same.

Magic Square With Sum 15
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Combinatorics was used to study these squares and the patterns that they hold within.

In the 12th century, there was an Indian mathematician named Bhaskara Acharya, whose work included descriptions of the integer coefficients of (a + b)^n, also called binomial coefficients. The study of these coefficients relied heavily on combinatorics.

Though combinatorics dates a long way back, it wasn't until the 17th century that it started to gain some attention. In the 17th century, French mathematicians Blaise Pascal and Pierre de Fermat developed probability theory, and with that came many combinatorial developments and results.

Pierre De Fermat
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Blaise Pascal
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