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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 and English Stackexchange version (both places where "mathematics" tags exist), 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.

nCr = 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:

$$\binom{n}{k}=\frac{n!}{k!(n-k)!}=\text{n nCrk}$$

nPr = permutations (EN)/variations (DE)

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:

$$\frac{n!}{(n-k)!} =\text{n nPr k}$$

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?

See also

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.

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    $\begingroup$ Aren't they called arrangements in French? This isn't the same as either German or English. And somebody should tell us what they're called in Russian. There's no obvious word for these things, so if they were developed separately in different languages, it shouldn't be surprising that they have different names. $\endgroup$ – Peter Shor Dec 2 '18 at 17:12
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    $\begingroup$ You might consider moving this to History of Science and Mathematics Stack Exchange. $\endgroup$ – Boshu Dec 2 '18 at 17:15
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    $\begingroup$ I mean hsm.stackexchange.com $\endgroup$ – Boshu Dec 2 '18 at 17:19
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    $\begingroup$ Differences between languages or between dialects do often represent different choices over time of which version of a word or phrase to keep (English and American being one example: each contains some words which the other abandoned a few centuries ago.) $\endgroup$ – timtfj Dec 2 '18 at 17:31
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    $\begingroup$ And in russian it's razmeshhenie, which also means arrangement, like in french and english. $\endgroup$ – Jean-Claude Arbaut Dec 2 '18 at 17:59
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$(\boldsymbol{1})\quad$ We call $\,P(n,k)\,$ k-permutations. Order matters, repetitions not allowed. The number of permutations of the n objects taken k at a time: $$P(n,k)=n(n-1)...(n-k+1)=\frac{n!}{(n-k)!}$$ So, these are ordered arrangements/selections/choices. We also use $\;_n P_k\;$notation.

$\quad$

$(\boldsymbol{2})\quad$ When we permute all objects we simply call them permutations and write $\,n!$

$\quad$

$(\boldsymbol{3})\quad$ If repetitions are allowed and order matters, we refer to such arrangements as permutations with repetitions or distinguishable permutations: $${{n}\choose{n_1, n_2, n_3…,n_p}} = \frac {n!}{n_1!\, n_2!\, n_3!…n_p!}$$

$$\quad$$

$(\boldsymbol{4})\quad$ When we have $n^k$ ordered arrangements, replacement allowed -- we call them permutations with replacements or k-tuples.

$$\quad$$

$(\boldsymbol{5})\quad$ When repetitions are not allowed and order doesn't matter, we call such arrangements combinations: ${{n}\choose{k}}\;$ or $\;_n C_k\;$ or $\;C(n,k).\;$ We choose n objects taken k at a time without regard to order. We read it "n choose k" and write: $${{n}\choose{k}}=\frac{n(n-1)...(n-k+1)}{k!}=\frac{n!}{k!\,(n-k)!}$$

$(\boldsymbol{6})\quad$ And finally, when we deal with unordered arrangements, repetitions allowed -- we call them combinations with repetitions: $${{n+k-1}\choose{k}}=\frac{(n+k-1)\cdot...\cdot n}{k!}=\frac{(n+k-1)!}{k!\,(n-1)!}$$

Please note it doesn't matter what these are called in German or French since each language has its own rules. An English manual of style is not applicable to other languages and vice versa; likewise you can't replace permutations or combinations with German variations in English. Yet we sometimes have different notations even within one language as authors may have their own preferences in terms of notation and terminology. There's nothing to worry about here.

Despite sounding a little pejorative to me, a German language speaker, it raises a question of whether such usage is really (internationally?) deprecated and considered outdated? -- No. That's more of a speculation on terminology used in other languages by some Wikipedia writers. See their editing history. Do not blindly trust something which is not a hard science in Wikipedia.

Do other languages possibly also name it differently? -- Yes. You can see it in the comments from people of other countries. There should be a lot of subtle examples. Let me make a rough guess based on "googling":

  1. Disposizioni semplici=Variation ohne Wiederholung=l'arrangement=k-permutations

  2. Arranjo com repetição=Variation mit Wiederholung=permutations with replacement.

So we should have been able to say these are just different technical terms -- but no -- it gets more complicated because in German terminology permutations are just a special kind of "variations". -- Yes, to some extent at least. Permutations is a broad term in English. What's more you can view these formulas from different "angles", e.g., combinations being just a special case of distinguishable permutations, permutations being just ordered combinations; and permutations with replacement can be called permutations with repetitions (it might create some confusion!) and so on.

The German flow chart with the German nomenclature? -- I checked it. It's really nice and logical. Quite commendable. I don't see how it may be inferior to any other nomenclature. It is rather on the contrary.

Nomenclatures, notations, and terminology differ from country to country. In biology, any species receives a binomial name (Latin name) and there's no ambiguity across the world about that species. In math we also have universal symbols and notations but they are not so rigid. You can find $cot^{-1}$, $arccot$, $arcctg $ used to denote the same and so forth. You can find in some countries analytic geometry is almost never part of calculus but always part of linear algebra. Sometimes you can find calculus being called mathematical analysis and being confused with analysis or real analysis. You may come across calques (or verbatim translations) of higher algebra, general algebra, etc. You may see how in English we coined words Calc I, II, III, IV as well as precalculus. Things are not clear cut, and there will be variations, just like the difference in meaning the word gift has in English and in German. While the word variation may have very similar meanings in English and German, there will also be differences, maybe subtle differences. And it is exactly the words with minor differences in meaning that cause most of the confusion. People expect them to be the same but they are not. One final example. We have books on vector calculus but it is a bit of a misnomer, as these books are just enhanced versions of Calculus III/IV. And it may have no bearing whatsoever on what might be the case in other languages.

CONCLUSION: Now we can answer the "title" question: Why are permutations P(n,r) called variations in languages other than English? -- That is simply not the case! While European languages may use math variations in a similar way, the usage will diverge to some extent.

IMPORTANT: Please note that not only technical terms are quite different from country to country but notations, too, may vary. Thus, in France, Russia, etc. permutations are often denoted $A^{k}_n$, and $C^{k}_n$ is used for combinations, which means the upper and bottom indexes are reversed. It may lead to mistakes in translation.

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    $\begingroup$ Great answer! 😃 $\endgroup$ – rugk Dec 18 '18 at 19:16
  • $\begingroup$ What would be the English notation for the $(2)$ permutations, $(3)$ permutations with repetitions or distinguishable permutations, $(4)$ permutations with replacements or k-tuples and $(6)$ combinations with repetitions? $\endgroup$ – Quaerendo Dec 1 '20 at 18:24

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