Let’s get a perm!
How many different ways can I arrange the pictures on my wall? How many different seating arrangements are possible in this office? How many different ways can I make this sandwich?
Maybe you’ve never been asked these questions (or asked them of yourself), but I have. Surprisingly, the number of possible arrangements in most circumstances is usually much higher than you may have thought. Trying to count them all on your own would probably take you way longer than it would take to make a sandwich, and honestly wouldn’t you rather be eating a delicious sandwich? I know I would.
Well luckily, there’s a very handy notion in Algebra called the Permutation of Numbers that can help you solve any of the above questions with ease. But first, a little background.
What’s a permutation you may be asking? Isn’t that when a newly permed head of hair gets wet and mutates into something freakishly hideous? Not at all. A permutation is basically a set of things (numbers, colors, or whatever) arranged in a particular order. Let’s say you had four colored blocks; red, yellow, blue and green. If you laid all the blocks together in that exact order, that would be considered 1 permutation of that set of blocks. But it’s certainly not the only combination you could make. By moving the last block (green) to the front of the line, you’ve now created a new permutation; green, red, yellow and blue. So how many different combinations could you make? Well let’s represent the number of colors by the letter X. Here’s the formula you would use:
So we have 4 colors right? Red, Yellow, Blue and Green. So let’s swap our number 4 in for X:
Well that doesn’t look so bad. Let’s simplify it a little bit:
and any number (not zero of course) multiplied by 1, equals itself. So our answer is 24. With 4 different colored blocks we can make a total of 24 different arrangements. Still don’t believe me? Well have a look-see:
(*edit* Thanks for the heads-up Lunatechie, the colors have now been fixed)
So what is our formula actually telling us to do? It’s telling us to count from our original number, down to 1, and then multiple all of those numbers together. With our colored blocks, we had 4 numbers, and we ended up multiplying the numbers 4, 3, 2 & 1. Another way to write this is simply “4!”. Putting the exclamation point behind the number 4 is telling us to multiply 4x3x2x1
Let’s try another example. Say there’s 8 different people in your office and you’re moving to a new building. There are 10 cubicles in the new office, which is great because you’re thinking of hiring 2 new employees next month because your business is taking off (congratulations!) Well it’s time to figure out who’s sitting where. How many options do you have? There’s 8 employees, but 10 is the key number here since there are 10 different cubicles available. Here’s our equation:
Say what???? That’s right buddy, with that amount of possibilities you may just want to let everyone pick their own seat.
Now what about that sandwich from before? Let’s say you have 2 slices of bread, 2 slices of turkey (or Smart turkey if you’re a vegetarian) and 3 slices of cheese because everything’s better with cheese. That’s 2 + 2 +3 for a total of 7. Try this one on your own.
Did you get it right? That’s 5,040 different ways to make your sandwich. Granted, it may not be the easiest to pick up (having the lunch-meat on the outside of the bread might seem strange, but try telling that to KFC).
That’s it, the permutation of numbers. Once you get the hang of this you’ll find yourself doing it for everything you see; your bookshelves, DVDs, cereal boxes in the pantry, the possibilities are endless. So enjoy your Permutations! (exclamation point intended)