Combinations And Permutations

With a new school year almost upon us, I've been reading up on the Math courses that I'll be tutoring shortly that I haven't done previously. One is a Grade 12 Data Management course that focuses on both probability and selection theory. The latter is largely concerned with combinations and permutations: knowing which is which, how to calculate each, and how to tell whether a particular scenario is one or the other.

For those who - like me - have forgotten some of this material, permutations are sets of items where the order of the items matters. In other words, in a permutation, a selection of {red, blue, green} is different than one of {blue, red, green} or {green, blue, red}. A combination, on the other hand, is a set selection where order is irrelevant (all similar sets containing the same items are considered to be the same set or outcome). As one of the websites humourously points out, we don't adhere very well to this Mathematical definition of "combination" in our every day use of the English language, or else those things that you use to safeguard your valuables in the fitness club would be called "permutation locks" (since the order of the numbers twirled on the lock are every bit as relevant as the numbers themselves, when it comes to getting it to open).

Anyway, half the battle in most questions of this type is figuring out whether you're dealing with a permutation or a combination. Naturally, you have to ask yourself, "Does the order of the items matter?" It can actually be quite challenging if you haven't done many examples of them already. Say you're trying to figure out how many different medal outcomes there are in a race of 10 runners where Gold, Silver and Bronze are awarded. In that case, you need to realize that order's important (Stephanie, Julie and Kait finishing 1st, 2nd and 3rd, respectively, is different than Julie, Kait and Stephanie doing so, for example). Therefore it's a permutation question.

Conversely, if 10 people at a party are vying for 3 door prizes, each of which is a $100 gift certificate to the Keg (and assuming that no one can win more than once), then order's immaterial. That's a straight-forward combination problem.

The similarity between the formulas for the two types probably adds to the confusion for some, as:

n P r (Permutations of n items when r are selected) = n! / (n - r)!

n C r (Combinations of n items when r are selected) = n! / r! (n - r)!

As you can see, the two formulas are almost identical. There's a reason for that, of course, as a combination is really a permutation in which you reduce your answer by all of the "duplicate" sets (those sets with the same items in different orders).

I knew all of this 25 to 30 years ago, of course... but I suspect that I'm going to understand it even better now, as I have to be able to explain it to others.