Introduction#

Combinatorics is the area of mathematics that deals with the art of counting.

Humans have been counting since the stone age. Most people learn how to count at a very young age. However, these aren’t good reasons to discount this subject as easy or trivial. Combinatorics is a rich subject with a lot of challenging and interesting problems, many of which are still open. There has been great progress in this field in the last half-century and it is still a site of active research.

Knowing how to count is a very useful skill and has applications in a wide variety of fields. Combinatorics has found uses in fields ranging from designing algorithms to calculating probabilities to statistical thermodynamics and more.

Combinatorics involves asking questions such as

• In how many ways can you arrange 10 books on a shelf?
• In how many ways can you write 42 as a sum of three positive numbers?
• How many different configurations does a Rubik’s cube have?

In this book, we will develop techniques that will help us count better and answer such questions.