In a new Pizza Hut commercial with the Muppets, Jessica Simpson pokes fun at her moron image by spouting off some math regarding the number of possible combinations for some new Pizza Hut product. Leaving aside the nonsense about probability which is thrown in apparentl because the copy writers thought it sounded “mathy,” the pertinent bits are:
Each pizza may have up to three toppings (so 0,1,2 or 3)
18 topping choices
According to JS, this means there are 6,321,000 possible pizza combinations. Is she correct? Assume no duplicate pizzas.
I don’t see how she could be correct. Adding ‘no topping’ into the topping choices to make it 19 (this takes care of pizzas with fewer than three toppings), we get 19^3, or 6859, as the number of different individual pizzas that are available, and 6869x6858x6857x6856*, or 2.21e15 as the number of pizza combinations. This is a lot more than 6 million.
*Not sure I’m doing that exactly right, but it’s not going to be out by so much as an order of magnitude, and I’ve got 9 of those to work with before JS is right.
I wonder if they really have 18 different toppings, or whether they are counting some preset combinations as a “topping”, for instance “Supreme” or “Meat Lovers”. You wouldn’t be able to combine that with two other options so it cuts the choices down a lot.
It occurs to me that by simply calculating pizzas as 19^3, I’ve included a lot of duplicates, since a pepperoni-pepper-mushroom is the same as a mushroom-pepperoni-pepper, etc. I’m too lazy to correct this error.
The information on number of toppings is from the commercial. The ad states 18 toppings. The commercial doesn’t indicate that specialty combos like Meat Lovers or Supreme are options, so assume they aren’t.
It’s been a long time since I did this sort of math, but I think the possible number of combinations of 18 things taken 4 at a time is:
C = 18!/4! * 14! which yields 3060 combinations
If you multiply that times 4 pizzas, you get only 12,240 combinations, but if you take 3060 things 4 at a time, then you have to calculate 3060! which my poor little calculator can’t do. Maybe someone with a calculator that handles extremely large numbers will come along.
The number of available pizzas is the sum of the number of pizzas with i toppings, where i ranges from 0 to 3. Since there are 18 toppings and order doesn’t matter, the number of pizzas with i toppings is 18!/(i!(18 - i)!), or [sub]18[/sub]C[sub]i[/sub].
So the total number of pizzas is [sub]18[/sub]C[sub]0[/sub] + [sub]18[/sub]C[sub]1[/sub] + [sub]18[/sub]C[sub]2[/sub] + [sub]18[/sub]C[sub]3[/sub].
So the total number of available pizzas is 988. If you do allow duplicate pizzas, the total number of combinations is 988[sup]4[/sup], which is 952,857,108,736. If not, the number is 988 * 987 * 986 * 985, which is 947,081,258,760. Either way, Ms. Simpson is off by a little bit.
Wow! What fun! I’d been meaning to sit down and figure this out since I first saw the commercial. It had been ages since I’d done any type of math like this, and I couldn’t remember the first thing about it. I had to rediscover the pattern and the formula on my own. In the process, I independently rediscovered Pascal’s Triangle! (Once I got that far, I cheated and looked up the formula for finding the value of a given cell, rather than filling in 18 rows by hand.) By that process I got 988 possible individual pizzas and 947,081,258,760 combinations of four, with no duplicates. I see on preview that ultrafilter got the same thing. How gratifying! At least I know I’m smarter than Jessica.
I agree the figure used in the commercials is incorrect. My question is where did they come up with that number? Did they do some sort of screwy back-assward calculation, or did the just pull a number out of their asses?
The assumptions that lead to the smallest number of combinations are:
No two pizzas the same
No two toppings on one pizza the same (which is unrealistic since I know people who like olive-olive-olive pizza.)
You have 18+1 choices for the first topping. In the case that you choose one of the 18, you then have 17+1 choices for the second topping. In the case that you choose 17, you have 16+1 choices for the last topping. Finally, if you ended up with two real toppings, you have 2! ways to have done that, and if you ended up with 3 real toppings, you have 3! ways to have done that. This means there are:
1 + 18(1 + 17*(1/2 + 16/6) ) = 988
possible pizzas. Since you’re getting four of them, there are
988987986*985/(4!) = 39,461,719,115
So even with these tight assumptions, JS still seems a bit short.
I was about to post this, then previewed and saw ultrafilter’s post. I’m really curious as to how his formula works.
This is what I got, using methods that seem logical to me:
3 toppings: (assuming there are 17 toppings) 17 * 16 * 15 = 4080
2 toppings: 17 * 16 = 272
1 topping: 17
0 toppings: 1
for a total of 4370 topping combinations. If we have no duplicate pizzas amongst our 4, we have 4370 * 4369 * 4368 * 4367 = 364,191,078,931,680. So Ms. Simpson is still way wrong. And she gets more wrong if you start allowing other toppings, like extra cheese, or double pepperoni.
Anyway, I’d love it if ultrafilter or alan smithee could tell me the flaw in my logic. (And I don’t mean that in a sarcastic way; I really want to know)
BTW, has anyone noticed that whenever advertizers make up a number to represent the possible combinations of whatever they’re selling, they always grossly underestimate. I think John Allen Paulos pointed this out first in his great book Innumeracy. I wonder why it is.
(BTW, I’ve always thought “pork topping” was an incredibly poor choice for naming a topping. It sounds like they’re trying to get away with something, like “cheese food” or “chocolate-flavored coating”.
I just imagine someone saying, “Well, the lawyers say we can’t really call this stuff pork or sausage. They said pork topping is acceptable, as long as we don’t imply its actually edible. That means we can’t suggest people put in on their pizza, but if they happen to see it on the menu and ask for it, we don’t have to say that its technically for topping road construction.”)