Factorials and Permutations: How Many Ways Can You Shuffle a Deck of 52 Cards?

The Mind-Blowing Scale of 52!

A standard deck contains 52 playing cards. How many distinct ways can you arrange them?

$$52! = 52 \times 51 \times 50 \times \dots \times 2 \times 1 \approx 8.0658175 \times 10^{67}$$

To put $8 \times 10^{67}$ into perspective: If every human who ever lived shuffled a deck once every second since the Big Bang 13.8 billion years ago, they would have explored less than an infinitesimal fraction of 1% of all possible permutations.

Permutations vs Combinations

• Permutations ($P(n, r)$): Order matters (e.g., race finish podium, password pin code).

$$P(n, r) = \frac{n!}{(n - r)!}$$

• Combinations ($C(n, r)$): Order does not matter (e.g., lottery tickets, committee selection).

$$C(n, r) = \frac{n!}{r!(n - r)!}$$

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