Loading Calculator...
Please wait a moment
Please wait a moment
Calculate factorials (n!) with complete expansion and step-by-step solutions. See how factorials grow and understand the calculation process.
Enter a whole number from 0 to 170
Enter a number to calculate its factorial
Type a non-negative integer (0, 1, 2, 3, ...). The calculator supports numbers up to 170 due to computational limits.
Click Calculate to see the factorial result, complete expansion, and step-by-step solution.
Review the expansion to see all the numbers being multiplied, and check the scientific notation for very large results.
n! = n × (n-1) × (n-2) × ... × 2 × 1
Factorial of n is the product of all positive integers from 1 to n.
0! = 1, 1! = 1
By definition, factorial of 0 and 1 both equal 1.
n! = n × (n-1)!
Example: 5! = 5 × 4! = 5 × 24 = 120
n! grows very rapidly
10! = 3.6M, 20! = 2.4×10^18, 100! ≈ 10^157
| n | n! | Expansion |
|---|---|---|
| 0 | 1 | 1 (by definition) |
| 1 | 1 | 1 |
| 2 | 2 | 2 × 1 |
| 3 | 6 | 3 × 2 × 1 |
| 4 | 24 | 4 × 3 × 2 × 1 |
| 5 | 120 | 5 × 4 × 3 × 2 × 1 |
| 6 | 720 | 6 × 5 × 4 × 3 × 2 × 1 |
| 7 | 5,040 | 7 × ... × 1 |
| 8 | 40,320 | 8 × ... × 1 |
| 9 | 362,880 | 9 × ... × 1 |
| 10 | 3,628,800 | 10 × ... × 1 |
A factorial, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It's used in combinatorics, probability, and many areas of mathematics.
By convention and mathematical consistency, 0! = 1. This makes formulas work correctly, especially in combinatorics where 'choosing 0 items from n items' has exactly 1 way (choosing nothing). It also maintains the recursive property: n! = n × (n-1)!.
No, factorials are only defined for non-negative integers (0, 1, 2, 3, ...). There's no standard definition for factorial of negative numbers in basic mathematics, though the gamma function extends factorial to other values.
Each factorial is the previous factorial multiplied by an increasingly larger number. For example, 10! = 3.6 million, but 20! = 2.4 quintillion (2.4×10^18). This explosive growth is faster than exponential growth.
Factorials are used in: permutations and combinations (counting arrangements), probability calculations, calculus (Taylor series), algebra (binomial theorem), and statistics. They answer questions like 'how many ways can you arrange n items?'
Most calculators can compute up to 170! ≈ 7.26×10^306, which is close to the maximum number that can be represented in standard double-precision floating-point format. Beyond this, you need special big number libraries.
Factorials are fundamental in counting and arranging objects. If you have n different items and want to know how many different ways you can arrange them, the answer is n!. This is why factorials appear throughout combinatorics and probability theory.
The rapid growth of factorials makes them both powerful and challenging. While 5! = 120 is manageable, 100! is a number with 158 digits! This exponential-like growth is why factorials are so useful in analyzing algorithms and computational complexity.