Key facts
- 5! = 120; 10! = 3,628,800; 20! = 2,432,902,008,176,640,000.
- n! counts arrangements: 5 books shelve in 120 orders.
- 0! = 1 by definition — there is exactly one way to arrange nothing.
- Factorials explode: 52! ≈ 8 × 1067 — every card shuffle in history is unique.
The factorial — written n! — multiplies every whole number from 1 up to n. This free factorial calculator computes n! instantly for any non-negative whole number, and explains where factorials appear: arrangements, probability and combinations.
Enter n and get the answer at once. Watch how fast it grows: 10! is already in the millions, and 20! has 19 digits.
How to use
- Enter n, a whole number (0 or positive).
- The factorial appears instantly.
- Compare sizes in the table below — growth is explosive.
- Use results in permutation (nPr) and combination (nCr) formulas.
Factorial definition
| Term | Meaning |
|---|---|
| n | A whole number, 0 or greater. |
| n! | The product of every integer from 1 to n. |
0! = 1 is a definition, not a calculation — it keeps the formulas for permutations and combinations working when nothing is chosen.
Worked examples
Arranging books
5! = 1 × 2 × 3 × 4 × 5 = 120.
Five different books can be ordered on a shelf in 120 ways — 5 choices for the first slot, 4 for the next, and so on.
Ten digits
10! = 3,628,800.
Ten distinct digits (0–9) form over 3.6 million orderings — which is why a 10-digit code is hard to guess but a 4-digit PIN (10,000 combos) is not.
Factorial values
| n | n! | n | n! |
|---|---|---|---|
| 0 | 1 | 11 | 39,916,800 |
| 1 | 1 | 12 | 479,001,600 |
| 2 | 2 | 13 | 6,227,020,800 |
| 3 | 6 | 14 | 87,178,291,200 |
| 4 | 24 | 15 | 1,307,674,368,000 |
| 5 | 120 | 16 | 20,922,789,888,000 |
| 6 | 720 | 17 | 355,687,428,096,000 |
| 7 | 5,040 | 18 | 6,402,373,705,728,000 |
| 8 | 40,320 | 19 | 121,645,100,408,832,000 |
| 9 | 362,880 | 20 | 2.43 × 1018 |
| 10 | 3,628,800 |
52! ≈ 8.07 × 1067 — more than the seconds since the Big Bang squared. Every properly shuffled deck is almost certainly a first in history.
Frequently asked questions
Why is 0! equal to 1?
By definition — it is the empty product, and it makes the permutation and combination formulas work when r = 0 or r = n.
Can I take the factorial of a negative number?
No — factorials are defined for whole numbers 0, 1, 2, … only. (The gamma function extends the idea to other numbers.)
What are factorials used for?
Counting arrangements (permutations: n!), choosing subsets (combinations: n! ÷ (r!(n−r)!)), probability, and series in calculus.
How fast do factorials grow?
Faster than exponentials: 10! is 3.6 million, 20! is 2.4 quintillion, 70! exceeds 10100 (a googol).
What is the difference between permutations and combinations?
Permutations count ordered arrangements (n! ÷ (n−r)!); combinations count unordered selections (divide again by r!).
Key takeaways
- n! = 1×2×…×n; 0! = 1 by definition.
- Factorials count arrangements — 5 books shelve 120 ways.
- Growth is explosive: 20! already has 19 digits.
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