How many?

Every now and again I come back to probability, which is a sort of hobby of mine. Probability maths is very much concerned with the number of different ways in which things can happen, so let's look at how we go about calculating this.
Imagine that you are a photographer, who has been engaged by a family of five (Dad, Mum and three children) to take their photograph. You turn up one afternoon, to be told that they want their photo taken on the back lawn, standing in a line, but they simply can’t agree on who should stand where. After listening to them arguing for five minutes or so, you rather rashly suggest that, if it helps, you are prepared to take as many photos as necessary to cover all the different ways in which they can stand. After all, you have a couple of 36 exposure films in your bag, and there’s a good hour and a half of daylight left - surely that should be enough? Well, let’s see -
The best way to approach this, is to imagine that there are five little stakes in the ground, numbered 1, 2, 3, 4 and 5, indicating the five possible positions for people to stand. OK then, any of the five of them can stand in position 1, so you have five ways of filling position 1. Now any of the remaining 4 can stand in position 2 - so four ways of filling position 2, but more to the point, for each of the 5 ways of filling position 1, you have 4 ways of filling position 2. So there are 5 x 4 = 20 different ways of filling the first two positions. This isn’t a particularly big number, so we can easily prove this by listing them. If D stands for Dad, M for Mum and 1, 2 and 3 represent the three children, then we can have:
DM D1 D2 D3 /MD M1 M2 M3 / 1D 1M 12 13 / 2D 2M 21 23 / 3D 3M 31 32
so there you are - 20 in all. Following the same logic, for each of these 20 arrangements, any of the remaining three members of the family can fill position 3, so that gives 20 x 3 = 60 ways of filling the first three positions, and then either of the remaining two can fill position 4, and whoever is left takes position 5. So in total then, there are 5 x 4 x 3 x 2 x 1 different ways in which they can arrange themselves - a total of 120. So in fact your two rolls of film aren’t going to be enough, and anyway you’re going to have to get a shift on to take 120 photos in an hour and a half!
But there's more, which we'll look at later.