Combinatorics

The Waffle House is bad at combinatorics. Last time I visited the Waffle House in Atlanta, I took a photo of their menu. This section of the menu shows the various ways that they can prepare your hashbrowns. Notice that there are three sizes (regular, large, and triple) and seven options (smothered, covered, chunked, topped, diced, peppered, and capped). That means that the number of possible ways of ordering your hashbrowns is 3 * 2^{7} = 384. However, as you can see in the photo, they claim that there are 1,572,864, or 3 * 2^{19} ways to prepare your hashbrowns. Where does this number come from? Are there “secret” hashbrown options that aren’t on the menu? 

Jessica Simpson is also bad at combinatorics.
Jessica Simpson, along with the Muppets, stars in a commercial for Pizza Hut’s new 4forAll Pizza. Here’s a description of the commercial taken from the Pizza Hut press release:
Now, let’s carefully consider the total number of options. Each of the four pizzas is allowed up to four toppings, and there are a total of 18 toppings to choose from. Assuming no repeated toppings, there are 18 choose 4, or 3060, possibilities for each individual pizza. Since there are four pizzas, the total number of overall options on the 4forAll is 3060 choose 4, or approximately 3.6 trillion. This is assuming that no two pizzas are identical and that no pizzas have repeated toppings (e.g. extra cheese, double pepperoni). If we discard these two assumptions the number of possible combinations is even greater! So although Jessica was technically correct when she stated that there were “more than six million topping options,” this is a pretty crappy estimate — it’s off by six orders of magnitude. This is particularly embarrassing given that the commercial’s main “joke” was Jessica’s “ironic mathematic acumen.” 
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Back when I posted this, several people wanted to know the exact answer to the 4forAll counting problem, since the original post only provided a lower bound. Unfortunately I was wrong about the topping rules: Pizza Hut actually allows only 3 toppings per pizza. But as it turns out, Jessica’s estimate is still way off. After a bit of debate, we agreed on the following solution.
S(n,r), the number of selections of r objects from a set of n, where order is not important and repeats are allowed, is given by
S(n,r) = C(n+r1,r)
Since you are allowed 03 toppings per pizza, the number of possible pizzas is
S(18,3) + S(18,2) + S(18,1) + S(18,0) = 1330
Then, since there we can select four pizzas from these options, the number of possible FourforAll configurations is
S(1330,4), or 1.3096×10^11.
Thanks to combinatorics fans Janna and Owen for contributing to this solution.
Comment by monzy — February 28, 2006 #
A friend and I also tried to work out the logic behind the Waffle House combinations number. I clearly remember that when we accounted for some fairly clear list of . . . condiments, or something like that, it worked out right.
We had no calculator, so she did the multiplication by hand on a napkin. It worked out.
But I don’t remember what the other options were. I just remember it was fairly logical and didn’t involve much in the way of random inclusion of ideas.
Comment by Randall Munroe — February 28, 2006 #
The Waffle House number must be including condiments, although I’m not sure how many they must be thinking of to spike the number by that much. Note, though, that they include picante sauce as a suggestion on the menu even though it is not an orderable option. This implies that the number of possible combinations is affected by items not on the menu. Maybe you can ask for a special menu listing every item in the Waffle House that could possibly be placed on top of hash browns. “A pancake, right on top? Sure, why not?”
Comment by Robert Flaxman — January 19, 2007 #
I’m now ready to confirm that you can reach that number when also accounting for the “default” condiments (including salt and pepper) contained withing the wire rack at the head of the typical Waffle House table. However, with the addition and removal of various condiments from time to time, you would have to get very lucky to land it correctly on 1,572,864. Also, they aren’t counting syrup, which they will give you if you are willing to also order pancakes, or are particularly flirtatious with your server.
My favorite recommended nonmenu item at Waffle House? StrawberryCherryVanillaLime Cokeade. It tastes exactly like those little wax soda bottles.
Comment by Bunsen — October 7, 2007 #