Unnecessary Maths: My Milkshakes Bring Over a Million Boys to the Yard

Canterbury is home to the Shake Shed, an awesome little shop that sells delicious milkshakes.  They will make you a shake with anything from chocolate chip cookies to sherbert lemon, to strawberry jam to Star mix – and you can have up to 3 flavours.  Not only that, they’ll do it with a smile!

One day I was walking past and saw a sign outside, boastfully promising that there’s over a million different milkshakes to choose from.  I wondered – is it actually over a million?

So I asked.  Unfortunately, this wasn’t something that the people working on the till at the time knew off the top of their head.  Also, they didn’t know how many different flavours they had.  Time to do things the hard way….

First things first, we need to know how many different flavours there are.  Lets get counting ….

OK, so, they said milkshakes – so I’m not counting anything hot chocolate, or anything else.  Just milkshake flavours.  And, by my reckoning, they have 182 different items that you can have in your milkshake (yes, I really did just count them).

So then, as we can see above you can have between 1 and 3 flavours in your milkshake, and it can be one of two sizes.  You also have the choice between Skinny, Standard, Creamy and Soya milk.

Lets start simply.  Lets say our total number of flavours is – there are possible combinations with one flavour – simple enough

There are then combinations of two flavours. However, that’s all possible combinations, including ones where both flavours are the same – so we minus , which is the same as the first figure (as both are the same, it effectively counts as the same as 1 flavour)

And then combinations of 3 flavours – again, we can minus as the number where all 3 are the same.  But then we also have   combinations where two of the 3 are the same – i.e. we count 2 of the 3 to be “one” flavour, and use the previous figure of how many 2 combination shakes and eliminate them.

Add them all together, you get: ,  let and it comes out to be 5,995,444.  Waaay over our over 1 million already.  But that’s not the end of the choices: we can also choose from 4 different milks: multiply that by 4 and you get

23,981,776 combinations.

Almost twenty four million possibilities. 48 million if you count regular and large as being different.  However, quite a few of them will be horrible, because they contain black jacks (I mean, really? Liquorice? And milk?)  But, I wonder, how many of them will contain one given flavour?

We can say in the case of 1 flavour shakes, only 1 out of will have blackjacks in it.  Easy enough.  In the case of 2 flavour shakes, there will be out of flavours which have blackjacks as one of the two flavours.  And then, finally, there will be out of .  Adding it all together, that means that will have a particular flavour in it.  Let again, and that gives us 33,307 combinations in which a particular flavour features.  Ew.  That’s a lot of manky blackjack milkshakes.  Or, looking at it positively, that’s a lot of potentially yummy banoffee pie milkshakes.

Lets throw in some more numbers here.  I’d say that the average diameter of a milkshake cup is about 8cm, and about 12cm tall.  And probably contains about 330ml of milkshake (at a guess).

If you were to put every possible milkshake combination in a line, the line would be long enough to go between London and Birmingham 10 times.  If it was a straight line, unrestricted by water, it’d get you about 1/3rd of the way into the Atlantic Ocean.  Likewise (if you have a few million milkshake floatation devices) the line would be able to go from Canterbury to Fes in Morocco.

If you piled them up one on top of another (as if you are trying to build some kind of milky space elevator), the top milkshake would be at an altitude of 2,877 km high – which is about between 6 and 10 times higher than the orbit of the International Space Station.

If you poured out all of the possible milkshakes into a bucket, it would weigh about 7,900 tonnes, and be enough fluid to fill 3 Olympic Sized Swimming Pools.

If it takes you about 1 minute to drink an entire milkshake, then it’d take you almost 63 years of solid drinking to get through every single combination.  That’s not including toilet breaks.  And, y’know, having a life (not that I can really talk, I’m spending my spare time doing maths for no reason.  At least I don’t need to pee the entire time).

 

And out of all those combinations, the only one for me is Banoffee Pie.  Yummm….

UPDATE!: I did a piece on CSR FM based on this post! Unnececary Maths – Milkshakes