permutations: converting sums to factorials

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(note: core question in bold-text below)

I'm starting to self-study probability for, hopefully, some professional certifications. The good news - the only time limits I have for learning this stuff are self imposed and eventually I'm hoping to get my head around continuous random variables and the like.

I'm in the first chapter of the book related to combinatorial analysis and have had to a lot of my own home-made testing of different equations and and assumptions (such as binomial theorem) in order to get a properly earth-bound sense of them.

My title question is really this:

I notice that any combination is given by the equation n!/((n-r)!r!), the (n!r) just being there to create an elimination in the numerator of multiplied numbers on the top end equal to the number of terms expressed by the factorial, for instance an n of 10 and r of 3 (ie. 10 possible choices, 3 need to be chosen, how many combinations?) would look like (10*9*8 )/(3*2*1)

The thing that I really want to understand however, when I try mapping these out in Excel to get a visual fix on this I end up with a different interpretation and it's one that I don't know how to put in fully legitimate mathematical terms because it has repeating summations - ie. for n of 8 and r of 2 it turns out to be the sum of 7 (28 combinations which visually works itself out as 7+6+5+4+3+2+1 with the other possibility almost behaving as a left-boundary).

The more I sort of fiddled with this I was able to come up with the assumption that the summary way of doing the same process goes something like this:

r-1 summations of (n-r+1) = n! / (n-r)!r!

I'm trying to visually understand, in the case of, for example, 8 possibilities, 2 chosen and no repeats, how (2-2+1) summations of (8-2+1) converts to terms such as 8*7/2.

Any suggestions?

I might have just answered my own question to some extent.

With n = 7 and r = 2

to label each r separately rsub1 and rsub2, rsub1 if first has 7 selections, rsub2 has 6, the division by 2 deletes the alternative set where rsub2 was first and rsub1 second. which would yield 42 results in total instead of 21.

In other words I need to keep in mind that my nifty fractal-ish charts I'm creating to keep track of connections already have a reduction coming from the deletion of permutations by order of selection.

Sounds like you have a good grasp on the basic formula for computing combinations.
A good way to explore this more deeply is to familiarize yourself with the binomial theorem, and explore its various representations, such as Pascal's triangle.

TY. The binomial theorem is pretty neat and it's in the same chapter of the probability book I'm studying.

The thing that's almost a trap with it and so many other similar equations is it almost seems like a one-trick pony so it's easy to sort of throw it to the side, at the same time I had my mind opened a bit by the next chapter which gets into summations of P(E)'s both inclusive and exclusive and there are proofs where they'll do little things like set x equal to -1 and y equal to 1, sure enough binomial theorem came right back in to play there in a way I never would have thought of. There's a pretty extensive stretch of theory problems at the end of each chapter and I think it would behoove me to work on those as well - mainly that all the theoretical language is still largely incoherent to me and since I'm going to see a lot more of it I should probably work on gaining a comfort level.

The binomial theorem keeps coming up. For instance, the binomial coefficient plays a role in the counting argument for the proof of Sylow's theorem in group theory.

Counting is very important in math, even at very abstract levels. So the "one-trick-pony" ends up performing again and again.

Getting comfortable with the language takes time and practice. Even after years of math, I still find I have to pay attention to the language, get used to whatever notation is being used in the material I'm working with. In many books I keep a sheet of paper with notes on the specific notation/language being used so I can quickly remind myself what the authors of that text chose.