Three shapes, three numbers
Before any cards are dealt, every hand has a fixed number of combinations, and there are only three answers to remember.
A pair is six because you are choosing two cards from four, and there are six ways to do that. Two different ranks is four suits times four suits, which is sixteen — of which four are suited and the other twelve are not.
That last line is worth sitting with. Offsuit hands are three times as common as suited ones, which is why an opponent turning up with suited connectors feels rarer than it is unfair.
Then the board takes cards away
Every card face up is a card nobody is holding. That is the whole idea, and it is called card removal or blocking.
There is an ace on that board, so only three aces remain. Choosing two from three gives three combinations, not six.
Half of the hands you were worried about stopped existing when the flop came down. Nothing about your opponent changed; the deck did.
Why this decides hands
Counting one hand is arithmetic. Counting two and comparing them is the part that wins pots.
On that same ace-high board, suppose you can only beat ace-king and you lose to aces. Aces are down to three combinations. Ace-king is now three remaining aces times four kings, which is twelve — and if you hold a king yourself, nine.
That 75% is the number that goes into pot odds. Combinations are how a read becomes a probability, and pot odds are what you do with it.
The two ways to get it wrong
Counting the full deck and forgetting the board
Answering six for pocket aces on an ace-high board is the single most common slip, and it doubles your estimate of the hands that beat you. Subtract the board before you count, not after.
Using the count for a different shape
Six, four and twelve are easy to swap under time pressure. Check whether you were asked for a pair, for suited, or for offsuit before you reach for a number.
What the count does not include
Combinations tell you how many ways a hand can exist, not how likely somebody is to be holding it. Nobody plays every hand they are dealt, so a range is combinations filtered by whether a person would actually have played that way. The counting comes first because it is the part that is objectively true; the filtering is judgement, and judgement applied to a wrong count is just a confident mistake.