How Fast Does n! Actually Grow? A Factorial Growth & Shuffling Guide
52! has 68 digits. That single fact explains why a shuffled deck of cards is, for all practical purposes, unique in the history of the universe. This guide walks through just how fast factorial growth outruns intuition, how to sanity-check a result with Stirling's approximation, how decimal factorial works through the Gamma function, and where a lesser-known cousin, the primorial, fits in.
Most people's intuition for "big numbers" comes from everyday counting, doubling, or maybe compound interest — all of which are exponential, growing by a fixed multiplier each step. Factorial is a different beast: at each step n, the multiplier itself keeps increasing, so n! outpaces any exponential a^n once n gets large enough. The clearest place to feel this is a deck of playing cards.
Why does a shuffled deck of cards feel "impossible" to repeat?
Short answer: A standard 52-card deck has 52! possible orderings — a number with 68 digits. Compared to that, the number of seconds since the Big Bang (roughly 4.3 x 10^17) or the estimated number of atoms in the observable universe (around 10^80) barely register as a fraction of the possibilities, which is why a well-shuffled deck is treated as effectively never-before-seen.
To put 52! in context: entering 52 into the "n!" tab of a factorial calculator returns a 68-digit exact integer, alongside its scientific-notation approximation (about 8.07 x 10^67). No physical shuffling process — human or machine — has ever come close to sampling a meaningful fraction of that space, and it never will. This is the same underlying math that powers cryptographic key spaces and lottery-odds calculations: once the input to a factorial reaches even modest double digits, the output stops being a "big number" in any everyday sense and becomes effectively astronomical.
Illustrative comparison: Suppose (hypothetically) every person on Earth shuffled a fresh deck once per second since the year 2000. That's roughly 8 billion people x 800 million seconds, or about 6.4 x 10^18 shuffles — a large number, but still a vanishing fraction of the 8 x 10^67 possible orderings. This example is purely illustrative, not a measured statistic.
How do you sanity-check a factorial result without redoing the whole multiplication?
Short answer: Use Stirling's approximation: n! is approximately equal to the square root of (2*pi*n) times (n/e) raised to the power n. It won't give you the exact digits, but it gives an estimated order of magnitude and digit count that a correct exact calculation should match closely.
This matters most when you're working with very large n and want a quick confidence check before trusting a multi-hundred-digit result. If a tool reports that 500! has, say, 1135 digits, Stirling's formula independently estimates almost the same digit count — and if the two ever disagreed by more than one digit, that would be a strong signal that something in the exact calculation (or in how the digit count was measured) had gone wrong.
Digit-count cross-check: exact result vs. Stirling's estimate (illustrative)
n
Exact digit count
Stirling estimate
50!
65
~65
100!
158
~158
500!
1135
~1134
1000!
2568
~2567
Another independent check, useful for a slightly different purpose, is the trailing-zero count: n! ends in exactly floor(n/5) + floor(n/25) + floor(n/125) + ... zeros (Legendre's formula). It doesn't validate the whole number, but it's a fast way to catch an obviously wrong result at a glance.
What does it even mean to take the factorial of a decimal?
Short answer: Ordinary factorial is only defined for non-negative integers, but the Gamma function extends the idea to any real (or complex) number via x! = Gamma(x+1). For example, 0.5! works out to approximately 0.8862, which equals the square root of pi divided by 2 — a genuinely irrational value, not a rounding artifact.
This isn't just a mathematical curiosity: the Gamma function shows up naturally in probability distributions (the Gamma and Beta distributions), in physics, and anywhere a formula built from factorials needs to be evaluated at a non-integer point. A calculator that supports this mode typically uses a numerical method like the Lanczos approximation to get several correct significant digits without symbolic calculus. One quirk worth knowing: the Gamma function has poles at zero and every negative integer, so factorial is undefined at those specific points even though it's defined everywhere else on the real line.
Worked example: (-0.5)! is not undefined, as one might guess — it equals the square root of pi, approximately 1.7725. This is a well-known special value of the Gamma function and a good one to test against if you want to confirm a decimal-factorial tool is implemented correctly.
What is a primorial, and how does it relate to factorial?
Short answer: A primorial, written n#, is the product of all prime numbers less than or equal to n — not all integers. For example, 10# = 2 x 3 x 5 x 7 = 210, since 2, 3, 5, and 7 are the only primes up to 10.
Primorial is easy to confuse with factorial because the notation and the "product of numbers up to n" framing look similar, but the two grow very differently. Because far fewer numbers get multiplied in (only the primes, which thin out as n grows, per the prime number theorem), a primorial is always dramatically smaller than the ordinary factorial of the same n. 10# = 210 versus 10! = 3,628,800 is a good illustration of the gap even at a small n.
Primorials matter in number theory: they appear in Euclid's classical proof that there are infinitely many primes (the argument constructs a number one more than a primorial), and in the study of prime gaps and primality-related conjectures. They're a genuinely distinct object from factorial's double/multi/subfactorial family — worth knowing about even though it isn't (yet) one of the modes in this tool's calculator. If you work with prime products often, it's the kind of companion calculation that could reasonably sit alongside the existing factorial variants as a future addition — for now, computing it by hand for small n (multiply the primes up to n) is straightforward using the prime list this tool's reference tables already imply.
Want to run these numbers yourself — exact n! up to 10000!, decimal (Gamma) factorial, double/multifactorial, subfactorial, permutations, and combinations, all free and instant?
Why does a shuffled deck of cards never repeat in practice?
A standard 52-card deck has 52! possible orderings, a number with 68 digits. Even if every shuffle in history had produced a genuinely new arrangement, the fraction of the total 52! possibilities that have ever been reached is effectively zero. That is why, in practice, a well-shuffled deck is treated as if it has never existed in that exact order before.
What is Stirling's approximation used for?
Stirling's approximation, n! is approximately the square root of 2*pi*n times (n/e) raised to n, estimates factorial without multiplying every integer from 1 to n. It is mainly useful as a sanity check: if a calculated exact digit count and Stirling's estimated digit count disagree by more than one, something in the calculation likely went wrong.
Can factorial be calculated for a number that is not a whole integer?
Yes, through the Gamma function, which extends factorial to real (and even complex) numbers using the definition x! = Gamma(x+1). For example, 0.5! is approximately 0.8862, equal to the square root of pi divided by 2. This lets you evaluate expressions like 4.5! that have no meaning under the plain integer definition of factorial.
What is a primorial and how is it different from a factorial?
A primorial, written n#, is the product of all prime numbers less than or equal to n, rather than all integers. For example, 10# = 2 x 3 x 5 x 7 = 210, since 2, 3, 5, and 7 are the primes up to 10. Primorials grow more slowly than factorials of the same n because far fewer numbers are multiplied in, and they show up in number theory topics like prime gaps and Euclid's proof of infinitely many primes.
Is there a quick way to sanity-check a factorial result without a calculator?
A rough check is to compare the digit count of your result against Stirling's approximation, since the two should be within a digit of each other for any reasonably large n. Another sanity check is the trailing-zero count: n! should end in exactly floor(n/5) + floor(n/25) + floor(n/125) + ... zeros, so an exact result that does not match this count almost certainly has an error somewhere.
A note on this guide: The shuffling and "people shuffling decks since 2000" comparisons above are illustrative and hypothetical, meant to build intuition about scale, not measured statistics. Numeric facts about factorial values, digit counts, and special Gamma-function values (e.g. (-0.5)! = square root of pi) are exact mathematical results. This article is for general educational reference and isn't professional, financial, or statistical advice.