10 Factorial — What is 10! ?

10! — Results
Exact value  =  3,628,800
Number of digits  =  7
Scientific notation  =  3.628800 × 106
Stirling's approximation  ≈  3.598696 × 106
n! ≈ √(2πn) · (n/e)ⁿ

Factorial neighbourhood of 10

The table below shows how quickly factorials grow around 10.

n n! Digits × factor
8 40,320 5 × 8
9 362,880 6 × 9
10 3,628,800 7 × 10
11 39,916,800 8 × 11
12 479,001,600 9 × 12

Calculate another factorial

Try: 0 · 5 · 10 · 20 · 52 · 100 · 170

Integers from 0 to 170 (171! overflows IEEE 754 double).

About 10!

10! = 3,628,800 — about 3.6 million. This is the number of permutations of 10 distinct objects, and close to 106.56.

Two amazing facts about 10!

1) 10! = 6! × 7!

First, factor out 7! from 10!:

10! = 7! × 8 × 9 × 10

It then suffices to show that 8 × 9 × 10 = 6!. Decompose each side via prime factorization:

  • 8 × 9 × 10  =  23 × 32 × (2 × 5)  =  24 × 32 × 5
  • 6! = 1 × 2 × 3 × 4 × 5 × 6  =  21 × 3 × 22 × 5 × (2 × 3)  =  24 × 32 × 5

Both sides equal 24 × 32 × 5 = 720  ✓, therefore:

10! = 7! × 6! = 6! × 7!  ✓

2) 10! = the exact number of seconds in 6 weeks

Count the seconds in 6 weeks step by step:

6 weeks× 7 days/week= 42 days
42 days× 24 hours/day= 1,008 hours
1,008 hours× 60 min/hour= 60,480 minutes
60,480 minutes× 60 sec/min= 3,628,800 seconds

6 × 7 × 24 × 60 × 60 = 3,628,800 = 10!  ✓

Connections to other branches of mathematics

Like all factorials, 10! appears in the binomial coefficient C(10, k) = 10! / (k! · (10−k)!) for any 0 ≤ k ≤ 10. This counts the number of k-element subsets of a 10-element set. The row 10 of Pascal's triangle sums to 210 = 1,024.

About factorials

What is a factorial?

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1 (the empty product).

How fast do factorials grow?

Factorials grow faster than any exponential function. While 2ⁿ doubles at each step, n! multiplies by n — an ever-increasing factor. By n = 100, we already have 100! ≈ 9.33 × 10157, and 170! reaches about 7.26 × 10306.

What is Stirling's approximation?

Stirling's formula provides a practical way to estimate large factorials: n! ≈ √(2πn) · (n/e)ⁿ. The relative error is already below 1% for n = 10 and below 0.1% for n = 100. It is widely used in combinatorics, statistical mechanics, and information theory.

Factorials in combinatorics

Factorials count permutations: n! is the number of ways to arrange n distinct objects in a row. They also appear in combinations C(n,k) = n! / (k!(n−k)!), in Taylor series (the n-th term is divided by n!), and in the Gamma function: Γ(n+1) = n!.

Why does 171! overflow?

IEEE 754 double-precision floats can represent values up to ≈ 1.8 × 10308. Since 170! ≈ 7.26 × 10306 is within range but 171! ≈ 1.24 × 10309 is not, 171! evaluates to Infinity in most languages using native floats.

Frequently asked questions

0! = 1 by convention. This follows from the definition of the empty product: the product of no numbers is the multiplicative identity, 1. It also ensures that C(n,0) = n!/(0!·n!) = 1 remains consistent for all n.
100! has exactly 158 digits. This is computed as ⌊log₁₀(100!)⌋ + 1 = ⌊∑ log₁₀(k) for k=1 to 100⌋ + 1 = 157 + 1 = 158.
Exactly 52! ≈ 8.07 × 1067 ways. This number is so astronomically large that every shuffle performed in all of human history is almost certainly unique.
This tool computes exact integer values up to 170! — the largest factorial that fits in an IEEE 754 double. 170! has 307 digits.
The digit count is ⌊log₁₀(n!)⌋ + 1. By logarithm properties, log₁₀(n!) = ∑ log₁₀(k) for k=1 to n. This is computable in O(n) without big-integer arithmetic.
The Gamma function Γ(z) generalises factorials to complex numbers: Γ(n+1) = n! for any non-negative integer n. Γ(1/2) = √π, and Γ(3/2) = √π/2.

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