The basics of counting

Last updated: 10 Jul 2007

Permutations and combinations are techniques for counting how many different orders or selections can be made from a set.

This sounds a little complicated so lets start with a simple example.

Simple counting selections example

At an restaurant there are 5 different main courses, lasagna, risotto, steak, trout and pizza and 3 different deserts, tiramisu, icecream and cheesecake.

There are clearly many different orders that you could make for one main course and one dessert. For example you could order the lasagna and then the tiramisu or the risotto and then the icecream.

But precisely how many different possible orders could you make for 1 main course and 1 dessert?

Well you have five possible options for the main course and for each of those main courses you can make 3 different choices of desert so there are 5 × 3 = 15 different orders you could make.

If you are not clear on this then try writing down all the possible orders starting with all the orders that can be made once you have chosen lasagna for your main course and then all those with risotto as the main course and so on.

General approach to counting selections

It should be easy for you to see that when there are multiple choices to be made the number of possible selections is multiplied at each stage.

So if there had been m main courses and d desserts in the first example there would have been a total of md possible orders that could be made.

A more complex counting example

An Italian restaurant has a choice of 4 starters, 5 main courses and 3 desserts on it's menu. How many different orders of 1 starter, 1 main course and 1 dessert can be made?

Following our general rule you can see that

I.e. there are 60 different orders that can be made.

So far so simple, right?

Here is where we start to make it a little more complicated.

In this example the choices of starter, main and dessert were entirely independent. I.e. the main you chose didn't make any difference to which dessert you could choose and visa versa.

However there are situations where these choices are not entirely independent and this is where permutations and combinations come in.

Next page: Permutations