Pringles ran a Super Bowl advert. In case you’re looking for ways to give Pringles more money, apparently you can buy several tubes of Pringles and mix the flavours. (Pringles are a type of food. Super Bowl is a kind of sport. None of that matters, what matters is…) The advert shows a man stacking three Pringles together and claims there are 318,000 possibilities.

Lucy (@honeypisquared) alerted me to this claim and some tweets by @bobloch trying to calculate the number. Bob noticed that the Pringles website has an interactive function to pointlessly simulate the stacking. This insists on precisely three Pringles being stacked together and allows a choice of 18 flavours (on a computer) or 17 flavours (mobile version, as far as I can tell). The website allows repetition (you can choose the same flavour multiple times). If there are 17 options per slot and three slots to fill, this means there are \( 17^3 = 4913 \) possibilities. If there are 18 flavours, this increases to \( 18^3 = 5832 \). Neither are anywhere near big enough.

So where does 318,000 come from? @jjc578 asked Pringles directly. Here’s their response.

So this says there are 25 flavours (weirdly, given how many the website shows) and we are allowed 2, 3 or 4 in a stack (weirdly, given the website insists on 3 only). The comments about not counting different orderings of the same Pringles suggests we are after combinations. Then the number of ways of choosing 2, 3 or 4 from 25 is:

\[ \binom{25}{2}+\binom{25}{3}+\binom{25}{4}=15250 \text{.} \]

Still not big enough.

Then @DavidKButlerUoA worked it out. If we are arranging 2, 3 or 4 Pringles in a stack, chosen without repetition from 25 where order *does* matter, then we use permutations rather than combinations and get this calculation:

\[ \frac{25!}{(25-2)!}+\frac{25!}{(25-3)!}+\frac{25!}{(25-4)!}=318000 \text{.} \]

That’s our magic number! It doesn’t match the Pringles advert, website or Twitter reply, but it’s nice that it does align to some piece of correct maths!

I’m increasingly convinced that the reply from Pringles is just a poorly worded sentence, by someone not used to constructing rigorous mathematical statements – and they did actually mean different orders of the same stack are different. Presumably the web gadget treats different orders as different stacks, so they’re obviously thinking that.

The tweet says “the math […] assumes the same flavors are not stacked in a different order” – this might mean, the math assumes that within a given set of flavours, you’re not allowed to stack them in a different order, as doing so would give you a different thing.

Also: If I put two Pringles in my mouth, I could tell you straight away which flavour was underneath as I can taste it on my tongue.