Combination Calculator
Calculate combinations C(n, r) — the number of ways to choose r items from n without regard to order.
Factorial Calculator
Calculate the factorial of any non-negative integer (n!). See step-by-step multiplication and related values.
Non-negative integer (0-170)
Second value for comparison
10!⧉
3,628,800
Product of 1 through 10
Number of Digits⧉
7
10! is a 7-digit number
Trailing Zeros⧉
2
Determined by factors of 5: floor(10/5^k)
Stirling's Approx⧉
3.598696e+6
sqrt(2*pi*n)*(n/e)^n -- error: 0.8296%
(10-1)!⧉
362,880
10! / 10 = 3.6288e+5
(10+1)!⧉
39,916,800
10! * 11
n!!⧉
3,840
Double factorial: product of same-parity ints
!10 (Derangements)⧉
1,334,961
Permutations where no element stays in place
Step-by-step expansion:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
Digit Magnitude
7 digits (vs 158 for 100!)
Comparison: 10! vs 12!
10! = 3.6288e+6
12! = 4.7900e+8
Ratio: 10! / 12! = 7.5758e-3
Factorial Reference Table
| n | n! | Digits | Trailing 0s |
|---|---|---|---|
| 0 | 1 | 1 | 0 |
| 1 | 1 | 1 | 0 |
| 2 | 2 | 1 | 0 |
| 3 | 6 | 1 | 0 |
| 4 | 24 | 2 | 0 |
| 5 | 120 | 3 | 1 |
| 6 | 720 | 3 | 1 |
| 7 | 5,040 | 4 | 1 |
| 8 | 40,320 | 5 | 1 |
| 9 | 362,880 | 6 | 1 |
| 10 | 3,628,800 | 7 | 2 |
| 11 | 39,916,800 | 8 | 2 |
| 12 | 479,001,600 | 9 | 2 |
| 13 | 6,227,020,800 | 10 | 2 |
| 14 | 87,178,291,200 | 11 | 2 |
| 15 | 1,307,674,368,000 | 13 | 3 |
| 16 | 20,922,789,888,000 | 14 | 3 |
| 17 | 355,687,428,096,000 | 15 | 3 |
| 18 | 6,402,373,705,728,000 | 16 | 3 |
| 19 | 121,645,100,408,832,000 | 18 | 3 |
| 20 | 2,432,902,008,176,640,000 | 19 | 4 |
Growth Visualization (log10 scale)
1!
1 digits
5!
3 digits
10!
7 digits
15!
13 digits
20!
19 digits
25!
26 digits
50!
65 digits
100!
158 digits
Planning notes, formulas, and examples
About the Factorial Calculator
The Factorial Calculator computes n! (n factorial) for any non-negative integer. The factorial of n is the product of all positive integers from 1 to n: n! = n × (n−1) × ... × 2 × 1.
Factorials grow extremely fast. While 10! = 3,628,800, by 20! the result exceeds 2.4 quintillion. Despite this explosive growth, factorials are fundamental — they appear in permutations, combinations, probability distributions, Taylor series, and countless mathematical formulas.
This calculator handles values up to 170! (the limit of JavaScript's floating-point precision) and shows the number of trailing zeros, which is determined by the number of times 5 divides into n!.
When This Page Helps
Factorials grow astronomically fast and are tedious to compute by hand. This calculator provides exact results (up to floating-point limits) and shows step-by-step multiplication.
How to Use the Inputs
- Enter a non-negative integer.
- View n! (the factorial).
- See the number of digits and trailing zeros.
- Compare with (n−1)! and (n+1)! for context.
- Results are exact up to 170! (JavaScript limit).
Formula used
n! = n × (n−1) × (n−2) × ... × 2 × 1
0! = 1 (by definition)
Trailing zeros = Σ floor(n/5^k) for k = 1, 2, ..Example Calculation
Result: 3,628,800
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800. It has 7 digits and 2 trailing zeros.
Tips & Best Practices
- 0! = 1 by mathematical convention (the empty product).
- n! grows faster than any exponential function.
- Stirling's approximation: n! ≈ √(2πn) × (n/e)^n for large n.
- Trailing zeros come from factors of 10 = 2×5; count factors of 5.
- JavaScript can represent up to 170! before reaching Infinity.
- The gamma function extends factorials to non-integers: Γ(n) = (n−1)!.
Factorial in Combinatorics
The number of permutations of n objects is n!. Combinations use factorials: C(n,r) = n!/(r!(n−r)!). Many probability distributions (binomial, Poisson) involve factorials.
Stirling's Approximation
For large n, n! ≈ √(2πn)(n/e)^n. This approximation is remarkably accurate even for moderate n and is essential in statistical mechanics and information theory.
Computational Considerations
For exact results beyond 170!, arbitrary-precision arithmetic (BigInt) is needed. Many programming languages provide big-integer libraries specifically for factorial-related computations.
Mastering this concept provides a strong foundation for advanced coursework in mathematics, statistics, and related quantitative disciplines. Understanding when and how to apply factorials is essential for solving real-world problems in probability and combinatorics.
Sources & Methodology
Last updated:
Frequently Asked Questions
n! (read "n factorial") is the product of all positive integers up to n. For example, 5! = 120. It counts the number of ways to arrange n distinct items in order.
By convention and the empty product rule. There is exactly one way to arrange zero items. It also makes formulas like C(n,0)=1 work correctly.
Extremely fast. 10! = 3.6 million, 20! = 2.4 quintillion, and 100! has 158 digits. Factorials grow faster than exponential functions.
Trailing zeros come from factors of 10 = 2×5. Since there are always more factors of 2 than 5, count the factors of 5: floor(n/5) + floor(n/25) + floor(n/125) + ...
The gamma function Γ(z) extends factorials to all complex numbers except negative integers. For positive integers, Γ(n) = (n−1)!. Γ(0.5) = √π.
n!! is the product of all integers from 1 to n with the same parity. 7!! = 7×5×3×1 = 105. It appears in combinatorics and certain integrals.
More in this topic
Permutation Calculator
Calculate permutations P(n, r) — the number of ordered arrangements of r items from n.