License Plate Letter Combinations

On the way to school on Thursday, Oliver and I saw a personalized Prince Edward Island license plate that was 8-letters long, the longest that will fit on a standard license plate. I seized this as a teachable moment, and for the rest of the walk to school we tried to figure out how many possible combinations of personalized license plates this would result in.

Things went horribly wrong.

We reasoned that of there are 36 possible characters that can go in each of 8 spaces, the total number of possible license plates would be 8 x 36, or 288.

But that didn’t make any sense: there are obviously more than 288 cars on PEI, and so there must be more than 288 license plates.

So last night we returned to the problem, and started with something simple: a 2-character license plate, with the range of A, B, C available.

Our earlier (wrong) calculation would come up with 2 x 3 = 6 possible combinations. But look:

AA
AB
AC
BA
BB
BC
CA
CB
CC

That’s 9 combinations, not 6 combinations, so obviously our original formula was wrong.

The source of our error: simply multiplying the number of positions by the number of possible characters works fine if each possible character can only be used a single time.

So if you’re arranging Peter, Bobby and Sue into two seats, your options are:

Peter - Bobby
Peter - Sue
Bobby - Peter
Bobby - Sue
Sue - Peter
Sue - Bobby

In that situation, where there’s only one Peter, one Bobby, and one Sue, then 2 x 3 = 6 calculates the number of possible seating combinations.

But for license plates, there’s no such restriction: each letter can be used any number of times.

And so the proper calculation for our simple test is 3² — three squared. Or 3 x 3 = 9.

Or, more generally characters^positions. So for our PEI example, it’s 36⁸.

Or 2,821,109,907,456 combinations.

Which is to say, two trillion eight hundred twenty-one billion one hundred nine million nine hundred seven thousand four hundred fifty-six.

More than 288, thank goodness.

(It’s not exactly 36⁸, of course: some personalized combinations are disallowed by regulation,  like “any messaging related to public or well-known figures, including members of the government and other dignitaries.”)