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:
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:
However, that's not the case, actually it calculates the "variation", as it's called in German:
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:
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:
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:
- Here are the formulas for the calculator keys
- Here is another overview about the subject as Germans see it.
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.
