Probability & Combinatorics
📋 What it is
Combinatorics counts how many combinations are possible — e.g. how many different 5-card hands exist.
🗣️ Coach says
Counting big possibilities has its own math. The number of different 5-card poker hands is 2,598,960 — found by combinatorics, not by listing them. The key idea: when order does NOT matter, you count combinations, not sequences.
🧠 Memory hook
Order matters → permutations. Order doesn’t → combinations. A hand is a COMBINATION (the order you got the cards doesn’t matter).
😂 Giggle
What did the magician say after a perfect card trick?
"That was suite!"
😲 Whoa!
There are 2,598,960 possible 5-card hands, but only 4 royal flushes — which is why one is so rare (about 1 in 649,740).
✅ Quick check: For a 5-card hand, does the ORDER you were dealt the cards matter?
Say your answer out loud first — then reveal.
No — a hand is a combination; same 5 cards = same hand regardless of deal order.
When order doesn’t matter you count combinations, which is why hands are counted that way.
🪄 Trick question: Are there more 5-card hands, or more ways to ORDER 5 chosen cards?
Careful — think it through, then reveal.
More orderings — each hand can be arranged 120 different ways, so orderings vastly outnumber hands.
5 cards arrange in 5! = 120 orders, so counting order inflates the total.
🧪 Try it! (2 minutes)
List every 2-card combo from just {A, K, Q}. You should find exactly 3 (AK, AQ, KQ) — order ignored.