Why are combinatorics used so frequently in the higher stakes to determine what decision to make?
For example, if you have AA and the board reads QJTr, some people are less likely to put the opponent on AK because the fact that hero has two aces reduces the amount of AK combos villain can have. Is this a correct application?

However, if villain shows through his betting patterns that he really is likely to have the nuts, i.e. c/c flop, c/r brick turn (since sets probably raise flop) then is one supposed to be more willing to use combinatorics or line reading to determine what decision to make, since they lead to opposite decisions?

Or are combinatorics just a tiebreaker sort of thing when situations are really marginal?

Combinatronics are used jointly with your other reads to make a decision. Generally what you use it for is when you narrow down their range based on what lines they took to a couple of possibilities that they would have played exactly the same way. You then can work out which hands they are more likely to have based upon the available cards.

Combinatorics are used to help convert line reading into an odds calculation.

This sums it up much better than my response.

I understand that they can influence the likelihood similar hands in a given range, but why is that even a legitimate way?
Isn't the likelihood that he gets dealt AK the same likelihood he gets dealt KQ in this situation? Or if not, then how is the small difference able to influence the much more direct way of interpreting bet sizes, other tells etc.

Can you give me an example of a useful application? (or use my example)

The likelihood that he gets dealt, for example, AsKs is the same as the likelihood he gets dealt KsQs - as long as you haven't seen the As, Ks or Qs, that is. Every individual exact hand is equally likely. "AK" is several hands - any Ace and any King put together. 4 * 4 = 16 of them.

When we find out that certain exact hands are impossible - because we hold one of the cards involved, or it's on the board, or (this judgement is never exact) it's not in his range - every exact hand that we can't rule out is still equally likely. DUCY?

It doesn't. We interpret stuff, and then we decide whether calling, raising or folding will give us the best EV. Combinatorics is part of the math for the EV calculations.

We interpret the bet size, stats, tells, whatever... to figure out what kinds of hands he can have. Combinatorics lets us figure out a complete list of exact hands he can have, grouped into equivalent classes. That is, instead of listing every possible AK, we just say "AK (however many combos)". Then we can check our chances against each kind of hand, and use the combos to get a weighted average that gives us our overall chance of winning (our "equity"). Finally we compare that to pot odds and implied odds

Bet sizes etc will only narrow your opponents range down so much. You use bet sizes, betting lines etc to narrow the range down. Once you have it narrowed down it is very rare that you can put your opponent on a single specific hand. So as poker is a game of incomplete information by using combinatronics you can then assign a reasonably accurate weighting to the likely different hands your opponent has. You have to assume your range is correct, so then you can say given this range, my opponent has a pair x% of the time, a straight y% and total air z%. Once you have x, y and z you can then mathematically solve what the correct action is. Combinatronics lets you mathematically solve a situation in poker.

It is different from hot - cold equity in that hot - cold equity tells you your pot equity against a range. However hot - cold equity only solves a situation where you and your opponent will both be all-in ie. if your opponent shoves and you are deciding whether or not to call. An example of a situation where combinatronics may be useful is if your opponent raises you all-in on the river and you know it is either a hand that totally crushes you or is a bluff. Based on the betting patterns so far you get the range of hands he would have reached this position with and count the combos of the hands that you think he would value raise and the hands that he would bluff raise. So you can work out that vil is beating me with a combos, I beat b combos, this means I am ahead c% of the time. Based on the raise size and the times I am ahead I can then solve if calling is correct or not. If you are relying totally on reads you would reach a point where you can see vil is bluffing or beating you but cannot determine which is more likely. Combinatronics will tell you which is the more likely.

Ask a live 1/2 player how they figure it out and they'll tell you they "just when he has it and when he doesn't". Combinatorics is a MUCH better way.