Permutation & Combination Calculator · Practical guide
How Many 5 Card Poker Hands Are Possible?
2,598,960 — and the combination formula shows exactly why.
At a glance
The Answer: 2,598,960
A poker hand is a combination: the order you receive the cards does not matter, only which five cards you hold. From a standard 52-card deck, the number of 5-card hands is C(52,5) = 52! ÷ (5! × 47!) = 2,598,960.
Interactive tool
Permutation & Combination Calculator
Enter your own values, then compare the result with the examples below.
Results
Formulas:
nPr = n! / (n - r)!
nCr = n! / (r! × (n - r)!)
The Math, Step by Step
The 47! cancels most of 52!, leaving five factors on top:
(52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1) = 311,875,200 ÷ 120 = 2,598,960
The numerator, 311,875,200, is P(52,5) — the number of ordered ways to deal 5 cards. Dividing by 5! = 120 removes the orderings, since each hand can be arranged 120 ways.
What That Means for Odds
| Outcome | Number of Ways | Odds Against |
|---|---|---|
| One specific exact hand | 1 | 1 in 2,598,960 |
| Royal flush (any suit) | 4 | 1 in 649,740 |
| Any 5-card hand | 2,598,960 | — |
There are only 4 royal flushes — one per suit — so the odds of being dealt one are 4 out of 2,598,960, or about 1 in 649,740. Every poker hand ranking's rarity comes from dividing its combination count by that same 2,598,960 total. The calculator above handles C(52,5) instantly — enter n = 52 and r = 5.
Frequently Asked Questions
Why is a poker hand a combination and not a permutation?
Because the order you receive the cards does not matter. Ace-king-queen-jack-ten of spades is the same royal flush no matter how the five cards are arranged in your hand.
How many royal flushes are possible?
Exactly four — one for each suit. That is why the odds are 4 in 2,598,960, about 1 in 649,740.
What is P(52,5), the ordered version?
52 × 51 × 50 × 49 × 48 = 311,875,200 ordered deals. Dividing by 5! = 120 gives the 2,598,960 unordered hands.
How many 7-card hands are there?
C(52,7) = 133,784,560 — the pool used when evaluating seven-card games like Texas Hold'em.