Picking and choosing

In a post last April, I described some of the mathematics behind combinatorics – the branch of mathematics concerned with counting all the different ways to do something. Often, this kind of maths appears in daily life – if you’re trying to work out how many combinations of something are possible, or how many different orders something could occur in.

This question also seems to occur to marketing executives on a regular basis – when they realise that their product can be sold in many different combinations to the consumer. It seems very exciting to marketing teams, in a way that it’s not necessarily that exciting to anyone else, that the number of possible combinations might be very large.

The problem comes when marketing teams try to calculate the number of possibilities – and even though the maths involved is fairly straightforward, fail to check their answer is actually correct (presumably most companies employ at least one mathematician that they could run it by?) This happens surprisingly often, leading to wildly inaccurate or confusing claims.

A few years ago, I experienced this first hand – when I checked into a hotel, to find they had a sign explaining that their breakfast buffet offered ‘Endless possibilities (well, 35 trillion to be exact)’. My initial curiosity (‘Why would you claim something is endless when it’s finite?’ ‘Is the number exactly 35 trillion?’ ‘How would you calculate that?’) turned to fury when I realised that the answers were, respectively, ‘Because we don’t know how maths works’, ‘Probably not’ and ‘We’re not telling you’.

On a breakfast buffet, you could consider each item as something you either have, or don’t have, on your breakfast. This means the number of possible combinations would be two to the power of the number of things on your buffet – for each, you choose one of ‘yes’ or ‘no’. For the number to be in the region of 35 trillion, a suitable number of breakfast buffet options should be 45 – since 245 = 35,184,372,088,832. This is, regardless of your level of precision, not in any way exactly 35 trillion (it’s out by a little over 184 billion), but this could explain where that number has come from.

In the wake of this realisation, I looked carefully at the buffet to see if there were exactly 45 different foods to choose from, but things quickly became unclear (Do you count different flavours of jam as different options? And is two sausages a different item to one sausage?). I also surmised that since a sizeable proportion of the possible combinations would be objectively horrible (with several other suggestions for horrible combinations contributed by my Twitter followers), this impressive number of possibilities doesn’t necessarily mean an impressive number of enjoyable breakfast combinations.

Part of the problem with these kinds of adverts is that it’s not always clear what counts as a combination. Last year the team at The Aperiodical wrote about a promotional Superbowl advert run by the makers of Pringles, in which they introduce the concept of stacking three different flavours of Pringles together and eating them all in one go – clearly a delicious thing to do – and claimed excitedly that there were 318,000 possibilities.

Several people on Twitter tried to work out where this oddly specific number came from, and realised there are a lot of considerations. How many flavours are you choosing from? How many Pringles go in a stack – does it have to be three, like in the advert, or can it be another number? And does it matter what order the flavours go in? (Presumably, you get a stronger hit of whichever one touches your tongue first, so it would be a different experience if you put a BBQ one underneath a Sour Cream & Onion).

One person took the step of directly contacting Pringles about this, and their social media team stepped up. Their reply, which answered all these questions, was clearly informed by someone having done some actual maths. With 25 Pringles flavours, in a stack of two, three, or four, and counting all the different orderings, you do get a total of 318,000. You need to first calculate the number of ways to pick 2 things from 25 – which is 25 choices for the first thing, times 24 for the second – then add this to the number of ways to pick 3 things (25 × 24 × 23) and the number of ways to pick 4 things (25 × 24 × 23 × 22).

Strictly, the wording of their tweet, in which they say the calculation “assumes the same flavors are not stacked in a different order”, could imply that the ordering is not important (so we’d need to divide each term by the number of ways to reorder, two, three or four things respectively), but if you bear in mind the ambiguity of written English and the difficulty of communicating knotty maths concepts like this, I’m happy to give them the benefit of the doubt.

It turns out this kind of maths is difficult – not necessarily because of the numbers, but because you have to know exactly what question you’re answering. In 2002, McDonald’s launched the McChoice menu in the UK – offering eight different items and claiming 40,312 different possible combinations. It’s impressive, but also somewhat unclear where this number comes from (and Twitter wouldn’t exist for another four years, so you couldn’t ask their social media team).

With 8 items, each of which you can either have or not have, there are only 28 = 256 different combinations, which is not nearly as many as they claim. With a little thought, it becomes apparent that they’ve calculated the number of different orderings you can put 8 items of food in (8! = 40,320), but then conscientiously excluded the 8 meals consisting of exactly one item, subtracting 8 to give 40,312, which is patently nonsense.

If this makes you angry, you might be pleased to hear that 154 brave souls complained to the Advertising Standards Agency and McDonald’s were taken to court to fight the claim – which McDonalds won after an appeal, by claiming that’s not what they meant, changing their story and convincing the ASA that they just meant there were a lot of options (and if you include different flavours of milkshake and other variants, there are actually a lot more combinations than 40,312). But this mixing of methods did lead to at least one new mathematical discovery – the McCombination numbers (of the form N! – N), which have yet to find an actual practical use.

While it’s nice to see maths being used, these claims are often actually a bit meaningless. Even if you have exactly 45 items on your breakfast buffet, the number of breakfasts you can have won’t be 245 – since you could have more than one of some items, and eat them in a different order (which might make for a different meal) – so any attempt to calculate the actual total is futile. You certainly can’t claim ‘to be exact’!

Maybe it’s best to follow the lead of one of my favourite combination miscalculations in advertising – the Rubik’s cube. In the early days, it was advertised as having ‘Over 3 billion combinations’ – a very impressive number, but in truth slightly shy of the actual 43,252,003,274,489,856,000 (43 trillion) combinations possible on a 3x3x3 Rubik’s cube.

Before mathematicians worked out the maths behind the cube’s complicated twists and turns, the exact number might not have been known, even to the engineer Rubik who designed the cube. But even though 3 billion is a massive underestimate, in combination with the word ‘Over’, the statement is strictly mathematically accurate! That’s what I like to see.

Posted by Katie Steckles

is a mathematician based in Manchester, who gives talks and workshops on different areas of maths. She finished her PhD in 2011, and since then has talked about maths in schools, at science festivals, on BBC radio, at music festivals, as part of theatre shows and on the internet. Katie writes blog posts and editorials for The Aperiodical, a semi-regular maths news site.

1. A few years ago, I experienced this first hand – when I checked into a hotel, to find they had a sign explaining that their breakfast buffet offered ‘Endless possibilities (well, 35 trillion to be exact)’. My initial curiosity (‘Why would you claim something is endless when it’s finite?’ ‘Is the number exactly 35 trillion?’ ‘How would you calculate that?’) turned to fury when I realised that the answers were, respectively, ‘Because we don’t know how maths works’, ‘Probably not’ and ‘We’re not telling you’.

So ist das Leben, nicht jeder ist mathematisch hinreichend gebildet, für solche Fragen – i.p. ‘Picking and choosing’ weiß der Schreiber dieser Zeilen gesondert zu ergänzen :

Wenn eine Wahl in einem n-fach offenen Wahlzusammenhang offen steht, bedeutet die positive Wahl des Einzelnen immer auch die negative andere Wahl anderer Entität.
Was interessant ist, Dr. Webbaer bringt an dieser Stelle gerne diese alt-chinesische Weisheit bei :
Der Meister bietet dem Schüler an zwischen dem Genuss unterschiedlicher Teigwaren zu wählen, deren Beschaffenheit, deren Inneres dem Schüler unbekannt ist.
Der Schüler greift dann ein wenig zögerlich – und wahlfrei – zu und der Meister fängt an darauf herumzureiten, warum die Wahl so und nicht anders ausgefallen ist.
Der Schüler sagt dann wahrheitsgemäß, dass er einfach so zugegriffen hätte, woraufhin der Meister sich aufplustert und seinem Schüler zu erklären weiß, dass eine Wahl immer auch die Abwahl von anderem bedeutet, denn sonst wäre sie keine, unabhängig vom unbekannten oder zumindest unverstandenen Inhalt.
Als der Schüler bei diesbezüglicher Erläuterung ein wenig zu nörgeln anfängt, wie dies ist dem Wesen eines aufgeweckten Schülers inne ist, fängt der Meister an zu schmatzen und sagt : “Du hättest dieses vorzügliche Teigröllchen wählen sollen!”

MFG – WB

2. My initial curiosity (‘Why would you claim something is endless when it’s finite?’ ‘Is the number exactly 35 trillion?’ ‘How would you calculate that?’) turned to fury when I realised that the answers were, respectively, ‘Because we don’t know how maths works’, ‘Probably not’ and ‘We’re not telling you’.

This lament reminds me of my math teacher, who once told me about an experience at a shopping mall where a shopper was missing elementary arithmetic skills.
The math teacher let it be known that this world is full of math idiots – and that this would not mean anything good at all.
And such a world is going to the dogs, my maths teacher was convinced of that. Because it wasn’t just the man at the cash register who lacked elementary math skills, it was also politicians and decision-makers.
How many evils of this world, how many wrong decisions were probably due to the missing math skills?
A lot, my math teacher made clear!

• So in etwa, Herr “Holzherr”, wobei Dr. W an dieser Stelle gerne noch kurz ergänzt, was Mathematik ist :
Mathematik kann gut als systematisierte Kunst des Lernens oder Fähigkeitslehre verstanden werden, wobei sie abgezogen ist vom weltlichen Gegenstand und insofern wiederverwendbar, vor allem auch lehrfähig, die Menge meinend.
Geredet wird dann von einer Formalwissenschaft

3. Citation from the above text (Picking and choosing)

With 25 Pringles flavours, in a stack of two, three, or four, and counting all the different orderings, you do get a total of 318,000. You need to first calculate the number of ways to pick 2 things from 25 – which is 25 choices for the first thing, times 24 for the second – then add this to the number of ways to pick 3 things (25 × 24 × 23) and the number of ways to pick 4 things (25 × 24 × 23 × 22).

Strictly, the wording of their tweet, in which they say the calculation “assumes the same flavors are not stacked in a different order”, could imply that the ordering is not important (so we’d need to divide each term by the number of ways to reorder, two, three or four things respectively

♨️This kind of combinatorics 🎲, namely by means of example and solution model (in this case “sequential choosing/picking”), appeals to me most.

✍️🙏 However, many people learn combinatorics 🎲 in school rather with another method, namely by dividing the combinatorial problems into 2 (or more) classes with 2 subclasses each. In the end, these students simply know that there are permutations and combinations with and without repetition and which formula to use for each of them. So they can solve combinatorial problems without understanding how to get to the formulas they apply.
In my opinion, such a teaching, which only offers formulas for conceptual problem classes, has failed. Because education means in my eyes, the ability of mental comprehension. And that means, in relation to mathematics, that one should be able to develop the solution model for a mathematical problem in such a way that one oneself and a listener, understands how it works.

4. Combinatorics 🎲 is in my eyes a mathematical field 🍓 that an interested layman (or student 🎓) can explore even without formulas. Many combinatorial problems (⁉️) can even be illustrated and solved with tree 🌳graphs, where each bifurcation 🌵 corresponds to a choice point, and where the branches represent the choices.
The tree model thus provides an intuitive 💡 approach to combinatorics. Instead of speaking of an intuitive approach to a mathematical field, one could also speak of a mathematical narrative 💃🎻 in analogy to the linguistic field. And that could be helpful for beginners in mathematics. Narratives in the linguistic field often contain recurring leitmotifs 🎶 and repetitive elements which, like refrains, appear again and again and with which readers 📖 are familiar 💞, even waiting for .

But what about matematics in this respect?

Math 🆚 Literature
My educated guess towards mathematics as a discipline of thinking and problem solving can be summarized as follows:
For Mathematicians mathematics is the art of proving, for all others, mathematics is the art of solving abstract riddles.

And it is true: Everyone needs basic mathematical knowledge, but only a few have direct access to mathematical questions – and even today, only a few have developed a basic mathematical understanding. This even applies to people who consider themselves educated 👩‍🎓👨‍🎓 and who can often talk animatedly about history, politics and Shakespeare’s plays, but who know nothing about mathematics, which they are even proud of towards their peers.

But why is a knowledge of Shakespeare so much easier to acquire than a knowledge of algebra and differential calculus? An indication of why this is so may be found in a recent study in the UK which compared privately reading primary school children with those who hardly or never read books 🚫 📖. The school success of bookworms was significantly higher. So reading alone makes you smarter. Yet reading is a pleasure for bookworms 📚 and not a duty.
Here lies the dog buried! Because there are no mathematical books that a normal child enjoys reading and is even looking forward to the next page and chapter.

Enjoy Math?
But how would it be, if this was not at all due to mathematics in and for itself, but only to the way mathematics has been presented to the pupils so far?

• The school success of bookworms was significantly higher. So reading alone makes you smarter. [Kommentatorenfreund “Martin Holzherr”]

Oder es ist andersherum, der Verständige liest mehr, weil er daran glaubt, dass nicht nur Eigenleistung relevant ist.
Shakespeare ist übrigens nicht so einfach zu verstehen, es kann nur leichter behauptet werden ihn verstanden zu haben, als dies bei der Mathematik der Fall ist.
Mathematik soll insofern als abstrakte, losgelöste & tautologische Fähigkeitslehre nicht überschätzt werden, wenn es das erkennende Subjekt oft mit Verhalten zu tun hat, die “wegen Komplexität” nicht mathematisierbar sind, nur sog. Szientisten meinen anders.
Wobei Dr. Webbaer gerade in der Stimmung ist Mathematik als großartig zu sehen, vergleichsweise.
Wie Kinder an die Mathematik herangeführt werden sollen, ist eine spannende Frage, Kinder lechzen oft nicht an Textaufgaben, sondern sind nicht selten in der Lage komplex zu abstrahieren, talentierte Kinder sind insofern mathematisch von Exkursen, die das Soziale meinen, sog. Textaufgaben bleiben gemeint, nicht angeregt.
Verweise auf Anwendungen helfen.
MFG – WB

5. Picking and Choosing: Is freedom of choice the freedom we mean?
The phrase “Endless Possibilities (well, 35 trillion to be (in)exact” suggests a great choice to the (prospective) customer.
And isn’t freedom of choice freedom at all? You might think so when you read the following introductory sentence (quote) in the German Wikipedia under Freiheit 🗽:
Freedom (Latin libertas) is usually understood as the possibility of being able to choose and decide between different options without compulsion.
In this sense, the buyers of a Ford Model T were free in the days of the first cars. Because they could choose any car color 🌈 they wanted. However, they could only buy a black ⚫ model, because there was no other colour 🚫🎨.
The “ibis Breakfast” (advertisment from above article), on the other hand, claims to have 35 trillion possibilities. Doesn’t that mean (almost) unlimited freedom?
Well, that also depends on what there is to choose from. I think an inmate 🃏 on death row would hardly feel freer if he could choose not only between 3 but between 3000 ways to die.

External and Internal Freedom
In philosophy, at least, freedom is something completely different from freedom of choice.