# Why are permutations ($_nP_r$) called differently in non-English languages (“variations” in German)?

First of all, you should be at least a little familiar with combinatorics to understand that question. Some often used calculator keys in stochastic are the nCr and nPr ones.

Edit: I've first asked this question for the German, English Stackexchange version (both places where "mathematics" tags exist) and usual Math Stackoverflow, but as it turned out this is likely also the case for all non-English languages and the community here may be better suited to answer the question, I've also posted it here. Despite that, below the German language is referred to as an example, where it is called differently.
Edit2: Also posted at History of Science and Mathematics and linguistics.

Edit3: In French this seems to be called "arrangement". And in Russian "razmeshhenie", "which also means arrangement".

# $$_nC_r$$ = combinations

nCr is quite obvious. The "C" stands for "combinations" (actually those without repetition) and this is how they are called in German and English. That is just the binomial coefficient:

$$\binom{n}{k}=\frac{n!}{k!(n-k)!}=_n\!\!C_k$$

# $$_nP_r$$ = permutations (English)/variations (German)

Keeping that knowledge in mind, as a German, you would assume nPr is for calculating the permutations (without repetition, again), i.e. just:

$$n!$$

However, that's not the case, actually it calculates the "variation", as it's called in German:

$$\frac{n!}{(n-k)!} =_n\!\!C_r$$

And it is true: Actually the "P" does stand for "permutation" in English. So the last formula is what they call "permutation".

## Just different names?

So we could say, these are just different names, but no, it gets more complicated, because – using the German terms here again – permutations are just a special kind of variations. Essentially, it's the last formula, where k=n, i.e. you choose all items and do not select a subset when arranging them.

Obviously the English mathematics do not use the term "permutations" for the specific version we name it in German, but for the general version. Essentially this leads to another problem, however, when we look at nPr with repetition. All examples before where without repetition, but you have formulas for the ones with repetition, too.

So the "permutation with repetition"/"Variation mit Wiederholung" and is easy to calculate, you just:

$$n^k$$

Wikipedia does not seem to want to acknowledge the English term for that saying they have "sometimes been referred to" in this way… (Or is this actually something different as the formula is k^n?)

Anyway, if we assume the term is used like that, we've got another way to have German "Permutationen" "with repetition". This time, however, as in the German definition of permutations we do not select items, we just have multiple of the same items. So e.g. you have r, s, …, t same elements in n elements you get a formula like that:

$$\frac{k!}{r!\cdot s! \cdots t!}$$

And this is what we call "Permutation mit Wiederholung" in German. But what term is then used in the English for this kind of "repetition"?

# Questions

So how did this inconsistent naming across languages happen? Is there any "correct" term or has one term been invented before another one, so someone adapted it wrong? Do other languages possibly also name it differently, i.e. is the German naming the exception or the English one? And what term is used for "Permutationen mit Wiederholung"/same elements in a set in English then?

If you need some more understanding:

Edit: I found something: The English Wikipedia describes the term "variations" as:

• Variations without repetition, an archaic term in combinatorics still commonly used by non-English authors for k-permutations of n
• Variations with repetition, an archaic term in combinatorics still commonly used by non-English authors for n-tuples

Despite that sounding a little pejorative to me as a German speaker, it raises the question of whether this is really (internationally?) deprecated/outdated? Or what term is supposed to be used? Also the relation to tuples, which are – I thought – just a different concept of a list of numbers, is not clear to me. After all, I could not found any of the formulas I've just mentioned in the linked article.

• Inconsistent meanings for literal translations is hardly limited to mathematics. British vs. USA biscuits and the French "monter" vs. "monter dans" are two well-known examples. Idioms make their way into math just as easily as into general language. – Carl Witthoft Dec 3 '18 at 20:42
• I think they are often called differently even in English. The terms "permutation" and "combination"; and the notations ${}_nC_k$ and ${}_nP_k$ are found only in certain textbooks (especially at high-school level or below), but are unknown otherwise. – Gerald Edgar Dec 4 '18 at 1:55
• @GeraldEdgar Maybe it's different in text books, but it is certainly written like that on all calculators I know. Internationally, AFAIK. – rugk Dec 5 '18 at 18:45

A place to start for questions of this sort is Jeff Miller's website Earliest Known Uses of Some of the Words of Mathematics. On permutations and combinations one of his sources is Smith, History Of Mathematics Vol II, p.28 (freely available), where we read:

"The first work of any extent that is devoted to the subject was Jacques Bernoulli's Ars Conjectandi. This work contains the essential part of the theory of combinations as known today. In it appears in print for the first time, with the present meaning, the word "permutation". For this concept Leibniz had used variationes and Wallis had adopted alternationes. The word "combination" was used in the present sense by both Pascal and Wallis. Leibniz used complexiones for the general term, reserving combinationes for groups of two elements and conternationes for groups of three, words which he generalized by writing con2natio, con3natio, and so on."

Miller marks the attribution to Ars Conjectandi (1713) as incorrect and gives two earlier sources for "permutation", based on OED:

"In 1678 Thomas Strode, A Short Treatise of the Combinations, Elections, Permutations & Composition of Quantities, has: “By Variations, permutation or changes of the Places of Quantities, I mean, how many several ways any given Number of Quantities may be changed.”

Lexicon Technicum, or an universal English dictionary of arts and sciences (1710) has: “Variation, or Permutation of Quantities, is the changing any number of given Quantities, with respect to their Places.”"

So the "inconsistency" was there from the start, as it usually is with ideas from multiple origins, rather than coming from a breakthrough by a single person. In such cases it is the eventual consistency, if it happens, that requires explanation, not the inconsistency. Mathematicians often stick to the use encountered in the native language textbooks, and those tend to emphasize what was introduced by their "own". The calculus notation story on the continent vs Britain is well known, see Was English mathematics behind Europe by many years because of Newton's notation? When people encounter several different words for the same term they tend to introduce some distinctions, and adapt each word to a subclass of the original meaning. And so on.

These issues are not settled by good reasons. There are no "correct" terms, and therefore, no wrong adaptations. This is how language is supposed to evolve. By the way, permutation (perestanovka) is also used in Russian, although more typically in algebraic rather than combinatorial contexts.

• "The calculus notation story on the continent vs Britain is well known." I am sorry, but I am not sure what you are referring, too. I guess it would help if you added a link or so. 😃 – rugk Dec 5 '18 at 18:53

"...what term is used for "Permutationen mit Wiederholung"/same elements in a set in English?"

The combinatorics expressions you're looking for in English are probably 'permutations/combinations with/without replacement' &c, rather than 'with/without repetition'. See this answer on the math stack exchange: https://math.stackexchange.com/questions/474741/formula-for-combinations-with-replacement .

Also you might see Sampling With Replacement and Sampling Without Replacement for other usage examples.

Of course, as in many other cases of technical translation, it's not guaranteed that the (nearly) equivalent phrases have exactly the same scope. I hope that helps.

• Replacement is generally used in contexts where you're talking about probability—e.g., sampling. But repetition is often used in contexts where you're not. For example, here. – Peter Shor Dec 3 '18 at 0:57
• @peter-shor Thanks, that looks like a helpful tech-linguistic point. – terry-s Dec 3 '18 at 18:00
• @PeterShor 'repetition' still sounds 'off' to me. Wiktionary seems wrong here. – Mitch Dec 5 '18 at 17:29
• @Mitch: Wiktionary isn't the only one to use it. See here and here and here and Ngrams. – Peter Shor Dec 5 '18 at 18:36
• @PeterShor Oh OK, that makes sense. Replacement with balls and urns but repetition with other things. But an expanded Ngrams search does lean towards doubt of Wiktionary, that is, that it should be 'selection with replacement'. – Mitch Dec 5 '18 at 18:57

This isn't an answer, but I think it's too long for a comment.

Sometimes the most literal translation of a technical term doesn't sound quite right in the language you're translating to. For example, Schrödinger used the term verschränken in German to describe particles related by Einstein's spooky-action-as-a distance, and then he used the word entangled in English, because a more literal translation of verschränken didn't have the right feel.

So to answer this properly, you might have to go back to the history of the words arrangement in French, k-permutation in English, Variation in German, find out when they were first used and who first used them, see whether they cited any papers in other languages, and try to decide whether a literal translation would have been perceived as incorrect for some reason. For example, I don't think variation would sound right in English, although permutation and arrangement would both work.