I spent nine years focused on mathematics, first as an undergrad, then graduate student, then professor. After that? Diapers, playdates, and the quest for publication. It's been a wild ride, and during this A to Z Blog Challenge, I'm posting Mad-Cool-Math Nuggets as a nod to those crazy, number-filled days . . .
Image courtesy: Sunchild57 Photography
C is for Counting.
There are lots of types of counting problems in discrete mathematics. Today I'll explain the difference between permutations and combinations. It's way more fun (and easy) than Calculus. Trust me.
Suppose you have 10 super favorite books. How many different ways could you arrange them on a bookshelf? Well, you could alphabetize by the author's last name, or alphabetize by the title, or arrange them in the order you read them. Lots of options, right?
Now let's get mathy. You have 10 choices to pick a book for, say, the leftmost position. Now there are 9 books left to choose from for the second slot. Then 8 books left to choose from for the third slot, and so on. So the answer is 10x9x8x7x6x5x4x3x2x1 = 10! = 3,628,800.
(Why do we multiply? Go here for the rule of product. What is that exclamation point for? It's short hand for factorial, the product of a number with all the natural numbers smaller than it down to 1. So 4! = 4x3x2x1)
Wow, there are over three million different ways to arrange 10 books! In this problem, order matters, and all those 3,628,800 options are called permutations.
Suppose order doesn't matter? Suppose you just want to pick 4 of those 10 books to take with you to the beach? This is a combination problem. The number of ways to pick 4 out of 10 objects turns out to be 210. I'll send you here if you really want to know the formula.
Fun fact: Combination locks are misnamed because order matters plenty with these things. If your "combination" is 2-3-4, dialing 3-4-2 won't work will it? They should be called permutation locks!
I still have nightmares that I'm in junior high and can't remember the permutation for my locker. How about you?