Factorial Calculator
n! = 1×2×3×...×n
Factorial of n (written n!) is the product of all positive integers from 1 to n: n! = n × (n−1) × ... × 2 × 1. Factorials appear in combinations, permutations, probability, and Taylor series expansions.
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Tip: Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ. For large n, this is very accurate. For n=10: exact = 3,628,800; Stirling gives 3,598,696 (0.83% error).
- 10! = 1 (by definition)
- 21! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120
- 3n! = n × (n-1)!
- 4C(n,k) = n! / (k! × (n-k)!) — factorials underlie combination formulas
5!=1205×4×3×2×1 = 120
10!=3,628,800About 3.6 million
20!=2.43 × 10¹⁸Over 2 quintillion
| n | n! | Approximate |
|---|---|---|
| 0 | 1 | — |
| 5 | 120 | ~10² |
| 10 | 3,628,800 | ~10⁶ |
| 15 | 1,307,674,368,000 | ~10¹² |
| 20 | 2.43 × 10¹⁸ | ~10¹⁸ |
| 52 | 8.07 × 10⁶⁷ | Shuffled deck orderings |
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Fun Fact
The number of ways to shuffle a standard 52-card deck is 52! ≈ 8 × 10⁶⁷. This number is so large that every time you shuffle a deck, you are almost certainly creating an ordering that has never existed before in human history.
References
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