How to acquire the variety of pairwise mixes of a set of variables?
I am attempting to recognize
The amount of mixes of examinations would certainly be there for instance, if
a
can take values from 1 to m
b
can take values from 1 to n
c
can take values from 1 to p
a
, b
and also c
can take m, n and also p distinctive values specifically. What are the complete variety of pairwise mixes feasible?
With a pairwise screening device that I am screening, I am obtaining 40 outcomes for m = n = p = 6. I am attempting to mathematically recognize just how I get 40 values.
If each parameter had $10$ selections you would certainly be examining $300$ vs $1000$ mixes, particularly hold $\rm a$ constant and also differ $\rm b,c$ via $10\cdot 10 = 100$ values. In a similar way hold, $\rm b$ constant ; after that $\rm c$. As the variety of variables $\rm k$ raises you improve financial savings, about $\rm (k N)^2$ vs. $\rm N^k$, where $\rm N =$ max domain name dimension. For QA objectives generally such harsh upper bounds are adequate. Do you have a desired application where you require something extra specific? If so probably you need to disclose some more information, as an example the circulation of the dimensions of the domain names, etc
EDIT : After assessing your most recent alteration, it shows up that the adhering to website might be of passion : Pairwise Testing, which describes numerous Taguchi methods such as those here. See additionally these web links to intros to combinatorial testing.
After reviewing this page, it appears that pairwise screening calls for a set of examination instances in which every set of values from any kind of 2 of the n groups takes place at the very least as soon as amongst the examination instance n  tuples. In the here and now instance, the trouble is to locate a marginal part of the 6x6x6 = 216 complete triples (a, b, c) such that

each set of values for an and also b
takes place at the very least as soon as, i.e. (a, b, *), 
each set of an and also c values takes place
at the very least as soon as, i.e. (a, *, c) 
each set of b and also c values takes place at the very least as soon as, i.e. (*, b, c)
Any part pleasing these needs have to contend the very least 36 components simply to please the (a, b, *) need. In the here and now instance I assume 36 examination instances are additionally enough, as in the adhering to set of triples:
(1, 1, 1), (1, 2, 2), (1, 3, 3), (1, 4, 4), (1, 5, 5), (1, 6, 6)
(2, 1, 6), (2, 2, 1), (2, 3, 2), (2, 4, 3), (2, 5, 4), (2, 6, 5)
(3, 1, 5), (3, 2, 6), (3, 3, 1), (3, 4, 2), (3, 5, 3), (3, 6, 4)
(4, 1, 4), (4, 2, 5), (4, 3, 6), (4, 4, 1), (4, 5, 2), (4, 6, 3)
(5, 1, 3), (5, 2, 4), (5, 3, 5), (5, 4, 6), (5, 5, 1), (5, 6, 2)
(6, 1, 2), (6, 2, 3), (6, 3, 4), (6, 4, 5), (6, 5, 6), (6, 6, 1)
In this instance each of the 3 sort of sets takes place as soon as and also just as soon as, i.e. there is no overlap. I do not assume this will certainly be feasible as a whole, so it could not constantly be very easy ahead up with marginal parts that cover all the instances.
Pairwise screening examinations for all feasible 2  means communications successfully   I offered a fast review below: https://cstheory.stackexchange.com/questions/891/
You are seeking toughness 2 covering arrays. In each set of columns every set of icons take place   this makes certain all 2  means communications are observed somehow. Below is a really straightforward instance of a covering array of toughness 2 with 2 columns:
11
12
21
22
12
What castel has actually attracted is basically the Latin square:
123456
612345
561234
456123
345612
234561
If you consider each access and also write the checklist (r, c, s), where r is the row index, c is the column index, and also s is the icon, you will certainly construct an orthogonal array (as shown listed below)   a covering array of toughness 2 with the minimum variety of rows (36 ).
111
122
133
...
661
In reality, Latin squares exist for all orders n. So if you have 3 columns (as an example 3 variables) and also n icons for each and every variable, after that you can constantly locate a toughness 2 covering array with n ^{2 } rows.
Several combinatorial layouts generate specifically reliable covering selections. Toughness 2 covering selections with greater than 3 columns and also n ^{2 } rows amount collections of mutually orthogonal Latin squares (the reference reveals the building and construction).
In your instance, if you have 40 outcomes, after that you are not making use of one of the most reliable covering array.
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