I think one of the first major "aha" moments for a poker player is that they grasp the concept of hand ranges and stop thinking about villain's holdings as a static single 2 card combination. However, i think that a lot of people ignore mathematics in the form of combinations to determine ranges, therefore don't understand with what frequency they have specific parts of their range.. Here is a hand rather crude theoretical hand example to illustrate:
150bb effective stacks, CO opens and you are on the btn and flat with AA. The flop comes J52r, and CO open jams. Lets say you have a reliable read that he is never bluffing here and only does this with overpairs and sets (assuming that we know that J5,J2,52 are not in his CO opening range). So we know his range here is consists of AA/KK/QQ/JJ/55/22. Ignoring AA cause there is no equity difference, we can see that we are beating KK/QQ and behind JJ/55/22. At first glance, we may think that we are actual behind more hands than we are ahead of, but using math of combinations, we can see that there are 12 combos of KK/QQ and 9 of JJ/55/22 meaning we are actually ahead of more of his range than we are behind.
so first, lets talk about how we actually calculate hand combinations, which ill break down into 3 parts using very simple mathematics:
1. Unpaired hands
to determine the number of combinations of unpaired hands multiply the number of the first card by the number of the 2nd card.
how many combos of KJ are there in the deck?
(4 Kings) x (4 Jacks) = 16 combos. Note there are always 16 combos of unpaired hands in the deck
how many combos of KJ are there on a KJJ flop?
(3 Kings) x (2 Jacks) = 6 combos.
2. Paired hands
To determine the number of combos, use the equation X(X-1)/2, where X is the number of the card remaining in the deck.
how many combos of KK are there?
(4 Kings)(4 Kings - 1)/2 = 4(3)/2 = 6 combos. Note there are always 6 combos of every pair in the deck.
How many combos of JJ are there on a J72 flop?
(3 jacks)(3 jacks -1)/2 = 3(2)/2)= 3 combos.
3. Suited hands
there isn't really math for this, its really simple, each card combo can have 4 suited combos out of the 16 total combos, obviously one of each suit. For example, there are 16 combos of AK, 12 of them are off suit and 4 of them are suited.
Now to the fun stuff, practical application:
You open 95hh on the BTN and the BB flats. The flop comes Ks9s5d. Villain checks, you bet and Villain raises.
Lets assume we know a decent amount about the villains game, so that we know he always 3bets JJ+/AK from the blinds, and only raises this flop with strong draws or sets. We can narrow his range down here to Axss, QJss, QTss, JTss, 99, 55. so we are crushed by sets and slightly ahead/flipping vs. his draws. Now understanding combos, we know that there are 10 combos of Axss (A2ss to AQss excluding A9ss), 1 combo each of QJss, JTss and QTss. And because we hold both a 9 and a 5, there are only 1 combo each of 99 and 55. So out of 15 combos of hands he can have, we are only don't have enough equity to get it in vs. 2 of those. And since his range is so heavily weighed toward draws, you can do stuff like call the raise and ship non spade turns without having to worry too much.
Now lets assume you have AsKc on this board now, and he raises, how does that change his range? Well his range is still theoretically the same (Axss, QJss, QTss, JTss, 99, 55), but using combinations, it is completely different. Now out of those hands, he can never have Axss and there are now 3 combinations each of 99/55 instead of one. So how he his range consists of 9 combos, 6 of which are sets and 3 of FD+gutter meaning you are now behind far more than you are flipping/slightly ahead.
This example just shows that hands that have almost the same absolute showdown value on that flop (TPTK vs. bottom 2 pair), can actually change villains range so much that one is an super easy call and one is a super easy fold (assuming he villain has negligible bluff frequency in both spots)
What you can take from this:
1. no one ever has anything. Out of like 1.3k combos possible in a deck, only a finite number of hands hit certain boards.
2. when you have something, its even more likely that no one has anything. The an obvious example is KK on KK6 flops where there are almost no combos of hands that can continue. Most hands are a lot less obvious though, but with a little intuition, its fairly easy to figure out.