CH 25: Your Order Will Take Forever
One interesting area of math deals with something called permutations. This differs from combinations in that order doesn't matter with a combination but order is everything in a permutation. Let’s start with a simple example, the letters J,A,Z. The permutations of the letters J, A and Z are JAZ, JZA, AJZ, AZJ, ZJA, ZAJ. This represents six ways to order three elements. Now it doesn't matter what the elements of the set are, they could be numbers, symbols, or people for that matter. So in our example the way we arrive mathematically at our six ways without having to write out every permutation by hand is to use what is called the factorial, denoted by the “!” symbol. Whenever you attach a “!” to a number it means that to arrive at the value, you have to multiply that number by each subsequent lower integer value until you get to 1. So for 3!, it really is just 3 x 2 x 1 = 6. This means then for a set of four unique letters, the number of permutations is 4! or 4 x 3 x 2 x 1 or 24. What you might be already realizing is that as you go up in the number of elements, the total number of permutations grows very fast.
Let’s consider a situation in which you have ten family members, arranging themselves in a line to take a group photograph. Like many families, an argument ensues and it is agreed that a photograph of every order of the family members should be taken to be fair. Assuming you have a fast camera and that everyone can move and take the next permutation of the family photograph every second, how long will it take to capture every way to take this picture?
Well, from the previous explanation you probably have surmised it is 10! seconds. How much time is this? Well, it is 3,628,800 seconds or 60,480 minutes or 1008 hours or 42 days exactly.
This is also assuming somehow that you never error in duplicating a previous permutation and have endless film. This might be a bit flabbergasting but no less a completely sound result from the realm of permutations.
Only 119 Pictures to go