How many five-card hands are there? The 2,598,960 explained

Why a 52-card deck makes exactly 2,598,960 different five-card hands, shown step by step with permutations, combinations and a simple divide-by-120 trick.

2026-11-03 · 5 min read

The short answer

There are exactly 2,598,960 different five-card hands in a standard 52-card deck. Every probability for a five-card hand starts from this number, because each hand is equally likely.

Step 1: count the ordered deals

Imagine dealing five cards one at a time and caring about the order. The first card can be any of 52. The second can be any of the remaining 51, then 50, 49 and 48.

52 × 51 × 50 × 49 × 48 = 311,875,200 ordered deals

Step 2: remove the order

A hand is a set of cards, so A♠ K♠ Q♠ J♠ 10♠ is the same hand however it was dealt. Any five cards can be arranged in 5 × 4 × 3 × 2 × 1 = 120 orders. Each hand has therefore been counted 120 times, so divide.

311,875,200 ÷ 120 = 2,598,960

The general formula

Choosing k items from n without caring about order is written C(n, k), read “n choose k”. It equals n! ÷ (k! × (n − k)!). For our case C(52, 5) = 2,598,960. You can try other sizes in the counting calculator on this site.

Why this number matters

To find the probability of any kind of hand, count the hands of that kind and divide by 2,598,960. For example 1,098,240 hands contain exactly one pair, so the probability is about 42.26%. The next post explains why that kind is so common.