Edit [
! I was (and still am) not up to speed on permutation and combination; I just wanted to flag the terms.
The original “permutation” illustration (below) is fine (except that I found an error in it). The “combination” illustration (below) was quite wrong, for the mathematical term; I have changed it.
In everyday English, we use “combination” for something like choosing a car — 3 engine options, 5 paint options and 2 body styles. That is {a meaning of the term “combination”} different from the mathematical one. (It is straightforward; 3*5*2=30.)
In maths, “permutation” and “combination” are particularly about choosing r options from among n. (Choosing the same option twice is excluded. Having repeats among the options is a special case; I am ignoring that here.)
• Permutation
We are choosing 2 options from among 5. Perhaps we are painting a model; we choose one colour for the body, and a second one for the trim.
The formula is nPr = n! / (n-r)!. The base concept is choosing 5 from among 5 — 5*4*3*2*1 = 5! [“5 factorial”]. That is the numerator. If we are choosing only 2, then we chop off the remaining options — …*3*2*1 — which we do by dividing by (n-r)!.
• Combination
One example here is choosing people for a team — 5 options, 3 choices. Again, the basis is 5*4*3, with …*2*1 removed by dividing by (n-r)!, as above. However, this gives us repeated selections of the same people; ABC,ACB,BAC,BCA,CAB,CBA are all the same option. For each such set, we have to reduce it to 1 option. We do this by [also] dividing by r! (here 3*2*1).
nCr = n! / (n-r!) / r!
]
[As ever, I really hate the fact that “the internet” insists that I could not possibly ever, ever, ever want more than one space.]
You want to know the terms “permutation” and “combination”.
Permutation is like this. For {ABCD} (n=4,r=4):
ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD …
Combination is like this. For {ABCD} (n=4,r=2):
AB AC AD BA BC BD CA CB CD DA DB DC.
These are mathematical terms, so for “permutation”… for (for instance) “AAAB”… with n=4, r=3, “AAB” would appear 6 times. [This [“6”] is offhand; I am not addressing the maths for duplicated options here.]
If you want to prohibit duplicates (e.g. AAB and AAB), you add the word “unique”, which is a strong word that means that, for a given thing, there is only one in the universe (of consideration). [I personally allow “very unique”, meaning not only unique but also strikingly different from everything else, but some people think that “very unique” is a contradiction.] …Or you can just say (add) that duplicates are prohibited.
“Adjacent” means “next to each other”.
Your formulation (“In how many words… always remain together?”) is actually pretty good. The difficulty is that, although it explains what you want, it is not strongly clear and precise, so the reader is a bit nervous about it. You want to use standard mathematical jargon here. (Do a search for puzzles with “permutation” or “combination”.)
[I (personally) am very strict with such things, so maybe the following is not representative [most people].]
The other issue is “word”. I [personally] would take that to mean an actual English word, even in a mathematics puzzle… although I would immediately think it odd. Indeed, I am inferring that here you do not mean “word” in that sense. [Actually, I think people do use “word” in this context meaning “any combination of letters”, but I certainly would not, myself. (Or I might use “ “word” ” (i.e. in scare quotes), if I was being lazy.)]
However, that problem is automatically solved if you use “permutation” or “combination”. (Failing that, I would say, “strings of letters”.)
So… “
How many permutations of the following letters — XXXX — are there… with the following constraint? [Generally: “The constraints are as follows.”]
• Vowels must always stand alone — i.e. be surrounded by consonants — except for one instance that can have 2 adjacent vowels.
”
It is more wordy, but that gives the reader confidence.
