What Is the Factorial of 100?
I walk through what 100 factorial actually equals, why the number is so unimaginably large, how mathematicians write it down, and where factorials show up...
Quick Answer
The factorial of 100, written 100!, is the product of every whole number from 1 to 100. The exact value is a 158 digit integer that begins with 9332621544394415... and ends in 24 trailing zeros. In scientific notation it is approximately 9.3326 times 10 to the power of 157.
To put that in perspective, the estimated number of atoms in the observable universe is around 10 to the power of 80. So 100 factorial is roughly 10 to the power of 77 times larger than the entire atomic count of the universe. Multiplication compounds shockingly fast.
You cannot compute 100! on most pocket calculators because it overflows a 64 bit floating point number. Python, Wolfram Alpha, and arbitrary precision libraries handle it without effort.
What a factorial actually is
A factorial is the product of all positive integers from 1 up to a chosen number n. The shorthand is n!, read as n factorial. So 5! is 5 times 4 times 3 times 2 times 1, which equals 120. The function grows extremely quickly because each new step multiplies by an ever larger integer.
- 5! equals 120
- 10! equals 3,628,800
- 20! equals 2,432,902,008,176,640,000
- 50! equals roughly 3.04 times 10 to the power of 64
- 100! equals roughly 9.33 times 10 to the power of 157
The exact value of 100 factorial
Here is the full integer, broken across lines for readability:
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
That is 158 digits. The 24 trailing zeros are not random; they come from the prime factorisation. Every pair of a 2 and a 5 in the factorisation produces a trailing zero, and 100! contains exactly 24 such pairs.
How to count the trailing zeros
You can compute the number of trailing zeros in n! without computing the full value, using a tidy formula attributed to Adrien Marie Legendre. Count how many multiples of 5, 25, 125, and so on are less than or equal to n, then add them up.
- Multiples of 5 in 1 to 100: 20
- Multiples of 25 in 1 to 100: 4
- Multiples of 125 in 1 to 100: 0
- Total trailing zeros: 20 + 4 = 24
This works because every multiple of 5 contributes at least one 5 to the factorisation, multiples of 25 contribute an extra 5, and so on. The number of 2s is always larger, so 5s are the limiting factor.
Why the number is so large
Linear growth doubles steadily. Exponential growth multiplies by a constant. Factorial growth multiplies by a number that itself keeps getting larger. At step 100 you are multiplying by 100, not by 2 or by e. That makes factorials the fastest growing function most people meet in school maths.
A quick comparison at n equals 100:
| Function at n = 100 | Approximate value |
|---|---|
| n (linear) | 100 |
| n squared (polynomial) | 10,000 |
| 2 to the n (exponential) | 1.27 x 10^30 |
| n factorial | 9.33 x 10^157 |
Stirling's approximation
Mathematicians rarely write factorials in full. They use Stirling's approximation, named for James Stirling who published it in 1730. The formula gives a very accurate estimate for large n:
n! is approximately the square root of 2 pi n, multiplied by n over e, all raised to the power n.
For n equals 100, Stirling's approximation gives roughly 9.3248 times 10 to the power of 157, which is within 0.1 percent of the true value. The relative error shrinks as n grows.
Where factorials show up
- Permutations. The number of distinct orderings of 100 items is exactly 100!. If you shuffle a 52 card deck, 52! is the number of possible arrangements, a number so large that almost every shuffle in history has produced a unique order.
- Combinations. The binomial coefficient, often written n choose k, uses factorials to count how many ways you can pick k items from n without caring about order.
- Probability. Many discrete probability calculations, including the Poisson and the binomial distributions, contain factorials.
- Series expansions. The Taylor series for e to the x, sine, and cosine all divide each term by a factorial.
- Algorithms. Brute force solutions to the travelling salesperson problem run in factorial time, which is why nobody uses brute force beyond about 12 cities.
How calculators and computers handle 100 factorial
A standard 64 bit floating point number can only represent values up to about 1.8 times 10 to the power of 308 with limited precision, and most pocket calculators cap out at 69! because 70! exceeds their display. To compute 100! exactly you need arbitrary precision arithmetic, which is built into Python's standard library (math.factorial), Java's BigInteger class, and tools like Wolfram Alpha.
Frequently asked questions
What is 0 factorial?
By definition, 0! equals 1. It looks odd, but it is the value that makes the combination and permutation formulas behave consistently. There is exactly one way to arrange zero items.
What is the largest factorial computers have calculated?
There is no theoretical limit. Researchers have computed factorials of numbers in the billions for distributed computing benchmarks. The bottleneck is memory and time, not mathematics.
Can factorials be negative or fractional?
Standard factorials only apply to non negative integers. The gamma function extends the idea to real and complex numbers, so the gamma of 5.5 gives a sensible answer even though 5.5! is not strictly defined.
Why does 100! end in so many zeros?
Each trailing zero comes from a pair of 2 and 5 in the prime factorisation. The 24 zeros in 100! reflect 24 such pairs, which Legendre's formula counts directly.