What is 6 factorial?
6! = 6×5×4×3×2×1 = 720.
10! = 3628800
Compute exact n! with BigInt—0! = 1, explosive growth, and the factorial backbone behind nPr and nCr counting formulas.
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n! is the product of every positive integer from 1 through n, with 0! = 1 by definition. So 5! = 5×4×3×2×1 = 120—the count of ways to order five distinct items.
10! = 3628800
Factorial counts how explosively ordered arrangements grow. Line up five books on a shelf: 5 choices for the first spot, 4 for the next, and so on—5! = 120 complete orderings.
Define 0! = 1 so combinatorics formulas do not break at empty sets. The permutation formula nPr = n!/(n−r)! and combination nCr = n!/(r!(n−r)!) both depend on factorials in numerator and denominator.
Growth is brutal: 10! = 3,628,800; 20! has 19 digits; 100! has 158 digits. EverydayTools uses BigInt for exact integers up to n = 500 in your browser—no scientific notation rounding on homework-sized n.
When a problem asks “how many ways” with ordering, think permutation first; when order is irrelevant, combinations cancel the r! reorderings inside nCr. This page gives the raw factorial building block.
Enter n → get exact n! locally. Chain to permutation or combination calculators when the problem specifies r selections.
Concise answers for common searches — definitions, steps, and comparisons.
6! = 6×5×4×3×2×1 = 720.
There is exactly one way to order zero items (the empty arrangement). Setting 0! = 1 keeps combinatorics formulas consistent.
Each step multiplies by a larger integer. 10! is 3.6 million; 20! is about 2.4×10¹⁸.
n! counts all orderings of n items. nPr = n!/(n−r)! counts orderings when only r positions are filled.
Multiply integers 1 through n using arbitrary-precision BigInt arithmetic. Result is exact for 0 ≤ n ≤ 500.
Formula
n! = ∏_{k=1}^{n} k. Stirling approximation (large n): n! ≈ √(2πn)(n/e)^n. Permutation: nPr = n!/(n−r)!. Combination: nCr = n!/(r!(n−r)!).Type n (0–500). Decimals and negatives trigger validation errors.
Full integer displays—copy for use in downstream formulas.
If selecting r from n, open permutation or combination tools instead of manual division chains.
Memorize 5! = 120 and 6! = 720 as anchors for mental estimates.
Input
n = 5Output
5! = 1205×4×3×2×1 = 120 distinct orderings of five items.
Input
n = 0Output
0! = 1Empty product convention keeps nCr defined when r = 0 or r = n.
Input
n = 1Output
1! = 1One item has exactly one ordering.
Input
n = 10Output
10! = 3,628,800Ten distinct books → 3.6 million shelf orders.
Input
n = 7 (for 7P3)Output
7! = 5,0407P3 = 7!/4! = 5040/24 = 210—factorial is the starting point.
Input
n = 15Output
15! = 1,307,674,368,000Already over a trillion—why counting problems need formulas, not brute lists.
Common real-world scenarios where this tool saves time.
Confirm 8! = 40,320 before dividing in a hand-computed 8P3 or 8C3 problem.
A deck of 10 labeled cards has 10! = 3,628,800 possible orderings—factorial gives the total shuffle space.
Brute-force traveling-salesman on n cities checks (n−1)! routes—see why n = 12 already means hundreds of millions.
Equally likely outcomes for ordering n items use n! in the denominator of each specific sequence probability.
Step-by-step chains that connect related tools for common tasks.
| n | n! | Digits |
|---|---|---|
| 0 | 1 | 1 |
| 5 | 120 | 3 |
| 8 | 40,320 | 5 |
| 12 | 479,001,600 | 9 |
| 15 | 1,307,674,368,000 | 13 |
| 20 | 2,432,902,008,176,640,000 | 19 |
| Question | Formula | 5 items, pick 3 |
|---|---|---|
| All orderings of n | n! | 5! = 120 |
| Ordered pick of r | nPr = n!/(n−r)! | 5P3 = 60 |
| Unordered pick of r | nCr = n!/(r!(n−r)!) | 5C3 = 10 |
Use dedicated permutation or combination calculators for nPr and nCr.
In nPr = n!/(n−r)!, expand only r factors: n×(n−1)×… instead of computing huge factorials.
Speeds multiple-choice combinatorics and sanity checks.
ln(n!) grows roughly n ln n − n—connects to logarithm calculator for magnitude estimates.
Full n! is rarely the final answer when choosing r < n items.
0! = 1 by definition. Without it, nCr formulas fail for edge selections.
n! counts all orderings. Committees and hands need nCr, which divides out r! duplicate orders.
Scientific notation loses digits. Use BigInt exact mode here for homework verification.
Standard n! applies to non-negative integers only. Gamma function extends elsewhere—not this tool.
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This tool caps at n = 500 for exact BigInt results. Beyond that, use computer algebra systems.
n!! multiplies every other integer down—different function, not computed here.
Shuffling 10 distinct cards has 10! equally likely orders—each specific order has probability 1/10!.
No. n! multiplies 1×2×…×n; n^n multiplies n copies of n. For n > 1, n^n grows faster eventually but differs at small n.
Choosing {A,B,C} is one combination but 3! = 6 permutations—division removes reordering duplicates.
Not in standard combinatorics. Gamma function generalizes to non-integers but not negative integers.
Subfactorial formulas subtract cases with fixed points using alternating sums over factorials—advanced counting built on n!.
n is processed with BigInt in your browser—never uploaded.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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