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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: Also posted on English, Mathematics, 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 over k) = n! / (k!(n-k)! = n nCr k)

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:

n! / (n-k)! = 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:

k! / {r! * s! * … * 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:


BTW, I am totally unsure whether to ask this in an English Stackexchange, Mathematics or German, but as I guess this is more a linguistic question about a specific feature in the German language (or about a missing one in the English one 😉), I've asked this here. Feel free to suggest more appropriate places…


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

  • 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.

  • Okay, so as it seems to affects all languages I've asked this on the English Stackexchange again: english.stackexchange.com/questions/475344/… You may close it here, unless it is really specific to the German language. – rugk Dec 2 '18 at 16:35
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    ruh, the general rule is to pick one site and ask there. Asking the same question on various sites is very much frowned upon. If you are unsure whether a question is a good fit, remember that each site has a) an "What topics can I ask about here?" page in the Help Center and b) a Meta site exactly for that kind of questions. (Does not apply if you ask about different aspects, obviously.) – Stephie Dec 2 '18 at 19:14
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    Warum fragst du das auf Englisch? Meinst du, eine Antwort bekommst du eher vom englischpräferierenden Teil der Leserschaft hier? - Ich habe nichts gegen englische Fragen; ich selbst schreibe hier mal so, mal so. Aber bei vorliegender sehr spezifischer Frage wundere ich mich schon. – Christian Geiselmann Dec 2 '18 at 23:37
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    The word »variations« is not a German word. It is English. Either you mean »Variationen« or »Varianten« (or what ever you have in your mind) or you are not talking about German language. – Hubert Schölnast Dec 3 '18 at 16:48
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    @HubertSchölnast I found no other way to triage where it may be best go/answered to. One (or, at least, I) cannot move questions from one platform to another. Also people repeatedly asked me to post it on another platform, so I did. For transparency and to make it easy to find a potential answer, I've all cross-linked them, though. – rugk Dec 5 '18 at 18:43
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In the traditional sense a permutation describes the rearangement of elements. That implies that you have to use every available element.

(2341) = (ABCD) -> (BCDA)

This is an example of one permutation. The concept "permutation" should not be confused with the formula for calculating the number of possible arangements.

(ABCD), (ABDC), (ACBD), (ACDB), (ADBC), (ADCB),
(BACD), (BADC), (BCAD), (BCDA), (BDAC), (BDCA),
(CABD), (CADB), (CBAD), (CBDA), (CDAB), (CDBA),
(DABC), (DACB), (DBAC), (DBCA), (DCAB), (DCBA)

Sometimes you are only interested in the number of possible arangements of a subset of the available elements. Example with two elements out of {A,B,C,D}.

(AB), (BA), (AC), (CA), (AD), (DA),
(BC), (CB), (BD), (DB),
(CD), (DC)

This case is called Variation in the german language and has a dedicated page. At the english wikipedia this case is dealt with in section Other uses of the term permutation

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