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Deck Shuffle Combinations Calculator

Enter the number of cards to find how many distinct shuffles are possible.

How it is calculated

The number of different orders a deck can be arranged in is the factorial n! = n × (n − 1) × … × 2 × 1, where n is the number of distinct cards: n choices for the top card, n − 1 for the next, and so on. Because the numbers grow too large to print, the tool adds up the natural logs of 2 to n, converts the total to log₁₀(n!), and splits it into an exponent and a mantissa: n! = mantissa × 10^exponent. The mantissa is shown with two decimals. For n up to 3 the exact whole number is shown, for example 6. The card count is rounded to a whole number and must be between 1 and 170; anything outside that range gives no result. All cards are assumed to be different, so repeated cards are not accounted for.

Worked example

Take a standard deck of 52 cards. The tool sums ln 2 + ln 3 + … + ln 52 and converts it to base 10: log₁₀(52!) ≈ 67.9066. The whole part, 67, is the exponent. The fractional part gives the mantissa: 10^0.9066 ≈ 8.07. The result is 8.07 × 10^67 distinct orderings. For comparison, a 10-card deck has only 3.63 × 10^6, and 13 cards already reach 6.23 × 10^9. Every extra card multiplies the total by the new card count.

Try this example →

Common mistakes

  • Reading 10^67 as 67 digits of ten. It means 1 followed by 67 zeros, then multiplied by 8.07.
  • Treating the answer as the number of different hands. This counts full orderings of the whole deck, not a hand of 5 or 7 cards.
  • Entering 52 for a deck with jokers or several decks combined. Add them into one count, but note that identical cards from several decks make some orderings the same, so the tool overstates the result.
  • Entering more than 170 cards. The tool stops there and returns no result.
  • Expecting the displayed two-decimal mantissa to be exact. 8.07 is a rounded value, and the true figure is slightly different.

About this calculator

The number of distinct ways to arrange a deck of n cards into a sequence is n factorial (n!) — n × (n−1) × (n−2) × ... × 1 — the standard combinatorial count of permutations of n distinct objects, since each of the n positions in the ordering removes one fewer choice for the next position. For a standard 52-card deck this number is astronomically large: 52! ≈ 8.0658 × 10^67, a figure so far beyond everyday intuition that it’s commonly used to illustrate just how effectively a well-shuffled deck guarantees a never-before-seen ordering — comparable in magnitude to estimates of the number of atoms in the observable galaxy.

This calculator computes n! for any chosen number of cards from 1 to 170, with the field pre-filled with the standard 52-card deck, and returns the result in scientific notation since the raw digit count quickly exceeds what’s practical to display in full. It’s used to illustrate basic combinatorics and probability — factorial growth, permutation counting — in statistics and math courses, and by card game enthusiasts curious about just how many distinct shuffles are actually possible.

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Comments (1)

  • Editorial team

    The answer changes enormously with the deck you actually use. A 40-card Spanish deck gives 8.16 × 10⁴⁷, a 52-card deck 8.07 × 10⁶⁷, and a 54-card deck with two jokers 2.31 × 10⁷¹. The tool assumes every card is distinct, so identical cards, such as a game with duplicated decks, would lower the real count. Which deck are you counting, and are you after the pure count or a comparison with something else?

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