Deck Shuffle Randomness Calculator
Every Well-Shuffled Deck Is a Number You've Never Seen Before
Shuffle a standard deck of cards properly and the exact order you get has almost certainly never existed before in the history of card playing. That's not an exaggeration — it's a direct consequence of how fast factorial growth outpaces anything intuitive, and this calculator shows the actual arithmetic behind that claim for any deck size.
The Formula
P(any single specific order) = 1 / n!
Entropy (bits) = log&sub2;(n!)
Entropy in bits is a measure of how much random information a shuffle actually needs to encode — a coin flip carries 1 bit, so a full 52-card shuffle carries roughly as much randomness as 225 independent coin flips.
How Fast Factorial Growth Gets Absurd
| Deck size | Total possible orders | Entropy (bits) |
|---|---|---|
| 13 (one suit) | 6,227,020,800 | 32.54 |
| 32 (Euchre deck) | 2.63 × 10³&sup5; | 117.66 |
| 52 (standard deck) | 8.07 × 10⁶⁽ | 225.58 |
For scale: 52! (about 8.07 × 10⁶⁽) is far larger than the estimated number of atoms making up planet Earth (roughly 10⁵⁰) — a properly shuffled deck reaches an order that has effectively never repeated.
Where This Gets Used
- Explaining shuffle quality — a "riffle shuffle" done only once or twice leaves the deck far from random; card magicians and casinos rely on this same math to define how many shuffles are actually needed.
- Teaching factorial growth and entropy — a standard deck is one of the most intuitive real-world examples of combinatorial explosion.
- Cryptography and RNG education — the entropy-in-bits framing connects directly to how random number generators are measured for unpredictability.
How to Use This Calculator
- Enter Deck Size — the number of cards being shuffled (up to 5,000).
- Select Calculate to see the total number of possible orderings, the probability of any one specific order, and the entropy in bits.
Related Calculations
See how those odds play out in an actual hand with the Poker Odds Calculator, or check drawing probabilities with the Solitaire Odds Calculator.