What is Combinatorics?

Combinatorics deals with mathematics surrounding counting. There are different "things" and methods to approaching this. There are also many applications of combinatorics in other areas of mathematics, such as graph theory, coding, probability, etc.

Factorials

We can count the number of orders in which something happens.
For example, if there are 3 students and 3 chairs, in how many different orders can the students sit on these chairs?

Solution: 6 differet possible orders

To simplify this, we can use factorials.

Try it yourself!

In how many different ways could 9 students sit on 9 chairs in a class?

Solution

Permutations

How do we approach the problem if there are not enough chairs?

We use for following formula

Important Permutation Reminders

Permutations are used if we care about the order
1! = 1
0! = 1

Try it yourself!

You forgot the code for your 4-digit lock. There are 10 digis (0-9) and each appears at most once. How many different ways can you try?

Solution

Combinations

Combinations are used when the order does not matter and you want to know how many ways there are to select a certain number of objects.

In a store, there are 5 different shirt colors: red, blue, green, yellow and black. You want to purchase 3 of them. How many ways are there to select 3 shirts out of the 5 that you like?

We use the following formula

Plug in the numbers...

Important Combination Reminders

Order does not matter
nCn = 1
nC0 = 1
nC1 = n
nCr = nC(n-r)

Try it yourself!

There are 10 students in your study group but you can only invite 5 to your party. How many different combinations of friends coul you invite?
(Note: use a pen and paper to help solve)

Solution

Test your knowledge