why divide lottery odds by combinations?

Because a lottery involves selection without replacement. Thus, the first number is picked from a group of 49 numbers, then the next number is picked from a group of 48 numbers, etc. Throwing dice, on the other hand, is simply a single random event.

I know it's a selection w/o replacement, but my brain refuses to see why 1 chance in 14 million and not 1 chance in 10 billion? Why exactly divide by 720.

The group of 48 numbers is the group of 49 minus the one that got extracted, so in every way, it's an independent event, meaning all the 48 numbers in it have the same chance regardless of which was picked from the 49 group to make it the 48 group.

From what I understand so far, the theory of getting the probability of winning 6/49 is logically flawed from start, adding the probability of getting all the permutations of the 6 matching numbers to win 6/49. So you have to divide by 720 to get the proper probability regardless of order, or permutations... but then, how do you modify the theory in the first place to sound right. I only understand this abstractly, I'm sure there is a deeper understanding about arriving at it.

The 10 billion gives you the odds for picking the six numbers in order. In other words, there's a 10,068,347,520 to 1 chance that I will pick the numbers 1,2,3,4,5, and 6, in that order, out of a hat containing slips of paper numbered 1-49. In a lottery, however, the order is unimportant. So if I pick 2,4,6,1,3, and 5 out of the hat, that would be the same result as if I picked 1,2,3,4,5, and 6 -- any ticket with those six numbers would be a winner. In order to make sure you're just talking about any series of six numbers, rather than a specific sequence of six numbers, you need to account for all the different permutations of any six numbers that might be drawn, and there are 720 (6!) different combinations of six numbers. That's why you divide by 720.

To make it more obvious, do a pick six with only six numbers. The number of permutations is 6! = 720, but since order doesn't matter, there is really one possibility. For each possible lottery result in a pick six, there are 720 ways to make it so you need to divide by 720 to find the number of unique combinations.

[EDIT] I think I understood joefromchicago's way of putting it, but there is more to it if you read below. I think I erred for abstracting even the actual numbered balls, like thinking of 49x48x... as distinct but variable at the same time picks... I think I probably invented a mental bias of my own ... or because it's friday night but that's how I build up my knowledge base.[/EDIT]

Yes, I understood that all along. What I don't understand is that by analogy, applying a proportion to a number is applying a fraction that represents that proportion, like 1/4 is clear I take 25% out of a number.

I acknowledge that order doesn' t matter, because I observed that, that I need to divide by the number of permutations.

I do not understand the inner working requirement however, or reasoning abstracting the fact I know the formula and observed the property that order doesn't matter.

It's as knowing what gravity does, when we don't know how it works...

It's because that lack of understanding I got no answers to this yet:
http://able2know.org/topic/175253-1

The understanding I'm talking of is the difference between formal and self education.

The key seems to stand in the fact that dividing n! by the number of combinations of numbers taken from it, practically unravels the basic partition w/o permutations from the n! as if "substracting" from the total possibilities, just in the way dividing by (n-r)! is "substracting" the repetitions out of all possibilities. So in other words I'm removing properties from all posibilities by dividing.

Like I said, the flaw in the wiki "lottery mathematics" is the way in which it is explained. They came up with permutations at first, or arangements how we call them. In my country they taught us that permutations are n!, n!/(n-r)! are arangements and n!/r!(n-r)! are combinations. Then they "substracted" the fact order is not important, because the number they came up with is describing just one arrangement, true. It would've been just as easy for me to assess that because a ticket has no order meaning and no repetition, I should apply the combinations formula directly.

However, in order to solve a complex problem, one need to be intimate with the meaning of the objects and operations meaning. For instance I know that in http://able2know.org/topic/175253-1 I need to apply (n+r-1)!/r!(n-1) to solve the probability of 4 out of 6, but not only I do not understand why I divide one thing by another, that won't help solve the problem because it's tied to the probability of guessing those 4 with another 5.

Then there's the another problem: take two dice, one is the lottery, one is the player. The chance the lottery and the player pick the same number is 1/6 * 1/6 = 1/36 not 1/6. If you don't like the player and lottery example, try two random generators, it's the same thing. So the probability of winning the lottery is not 1:14 million. That's just the probability of a particular outcome of the lottery alone it would seem. Dr. Math says it's not though no mathematical demonstration is given. Statistically I haven't tried for 6/49 probability, though all the other distributions seem to model the statistical reality pretty close, I would say the probability for winning 6/49 is lower than 14 million, speaking from experience. The math just doesn't make sense.

http://able2know.org/topic/175261-1

"It would've been just as easy for me to assess that because a ticket has no order meaning and no repetition, I should apply the combinations formula directly."

Bingo! They just took the long way. Dividing arrangements by the number of permutations of the arrangements yields combinations.

Yes, that wasn't the best approach for me

What I was getting at also, is that perhaps there are other types of intelligence to get to that answer. However to quote Roger Bacon:
"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."

Side note. Basicly there are at least as many types of intelligence as there are sciences and interdisciplinary domains, because intelligence is comprised of the notions, combinations of notions and methods that combine these notions at least. They are not all mathematical but I got to say there are very few others if I consider logic part of mathematics. For instance there are musical harmonics, some are abstracted experiences that yield general rules, emotional intelligence that abstract the solution directly from the ability of neurons to solve problems. Most of these side notions are subject to cognitive bias unfortunately. What I learned by experience with this is just how big is the importance of mathematics. It is truly interwoven with everything.
For example there are organic ways to understand and get to these results but they are impractical as it takes too many neurons I suppose to analyse combinations synthetically, logically and by other means, so I give up to pure mathematical abstraction and keep the others as metaphysics