This post is in the ‘Mathematics for Everyone’ category. These posts will be short snippets and will, I hope, increase slowly in complexity.
Suppose that you are going out to the cinema tonight and that you do not know what to wear. You have narrowed down your choice considerably but you cannot decide which of three different tops and which pair of two different pairs of jeans you would like to wear. You would wear any top with any pair of jeans, but you need to choose which top and which pair of jeans. How many different complete outfits are you choosing from?
Six. There are six different outfits available to you. As I said in the penultimate sentence of the previous paragraph, your choice of top is does not depend on your choice of jeans. So, for each top there are two outfits – one for each pair of jeans. That is two complete outfits per top. To labour this point even further, here is a colourful picture:
The reason why this picture is relevant is because each dot represents an outfit: Describing a particular outfit is exactly the same as pointing to a dot on the grid. Choosing your top is choosing which column your dot is in. Once you choose your top, i.e. once you choose a particular column, you still have to choose your jeans, i.e. a particular row. Now it is very easy to see that there are six outfits because we can count the dots. Even more crucially, there is a particularly nice way of counting them, which is to multiply the lengths of the sides together, i.e. to compute two times three.
Suppose now that we mix things up a bit and say you must also decide which of ten funky hats to wear. Suppose also, in the same way as before, that you do not mind which hat you wear with which top and with which jeans. How many outfits are there now? While it is considerably more difficult to draw a useful picture, the problem has not gotten significantly harder. The answer is sixty because sixty is equal to two times three times ten.