Factorial Calculator
Calculate the factorial of any number up to 170.
What Is the Factorial Calculator?
A factorial calculator computes n factorial, the product of all whole numbers from one through n, for any input your machine can handle. Factorials count arrangements: five people can line up in 5 factorial ways, a deck of cards shuffles into 52 factorial orders, and probability formulas across statistics lean on factorials constantly. Growth is the story here, since values explode beyond astronomical within a couple dozen steps; 70 factorial already exceeds the number of atoms in the observable universe. This tool returns exact integer results without floating point drift, handles big inputs gracefully, and shows the full multiplication chain on request.
Key Statistics
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120
Value of 5 factorial, the number of ways to arrange five distinct items
Source: Combinatorics definition
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About 8 times 10 to the 67th
Magnitude of 52 factorial, the possible orderings of a card deck
Source: Standard combinatorics result
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1
Both 0 factorial and 1 factorial equal one by definition
Source: Mathematical convention
The Formula
n! = n x (n-1) x (n-2) x ... x 2 x 1. By definition, 0! = 1.
Worked Examples
5! equals 5 times 4 times 3 times 2 times 1, or 120 ways to arrange 5 books on a shelf. 10! comes to 3,628,800, showing how quickly factorials grow. 7! is 5,040 and 12! surpasses 479 million, making these calculations essential for probability and counting problems in statistics.
Arranging books on a shelf
- You have 5 different books
- First slot has 5 choices, then 4, then 3, then 2, then 1
- Multiply: 5 times 4 times 3 times 2 times 1
There are 120 possible arrangements.
Real World Use Cases
Probability problems
Evaluate permutation and combination formulas that divide factorials.
Combinatorics coursework
Verify counting arguments in discrete mathematics assignments.
Puzzle analysis
Quantify how many states games, locks, and arrangements truly have.
Expert Tips
- ✓ Cancel factorials before multiplying when formulas contain ratios like 10 factorial over 7 factorial.
- ✓ Zero factorial equals one by convention, which keeps many combinatorial formulas consistent.
- ✓ For very large n, Stirling's approximation estimates magnitude without computing exact products.
- ✓ Factorials overflow standard floating point around 170, so exact integer arithmetic matters for big inputs.
Frequently Asked Questions
What is a factorial? ▼
A factorial, written with an exclamation mark like 5!, is the product of all positive integers from 1 up to that number. So 5! is 5 times 4 times 3 times 2 times 1, which equals 120.
What is 0! ? ▼
By convention, 0! equals 1. This keeps formulas working cleanly, especially in combinations and permutations where an empty set has exactly one way to arrange it.
How do I calculate a factorial? ▼
Multiply the number by every positive integer below it down to 1. For 6!, multiply 6 times 5 times 4 times 3 times 2 times 1 to get 720. This calculator handles it instantly.
What are factorials used for? ▼
Factorials count arrangements and selections, like how many ways to line up a group or choose a subset. They appear in permutations, combinations, probability, and series like the one that defines e.
How big can factorials get? ▼
They grow extremely fast. The number 10! is already 3,628,800, and 20! is about 2.43 followed by 18 zeros. Beyond about 25, the results get too large for standard calculators and need special handling.
What is the difference between n! and n!! ? ▼
A double factorial skips every other number. The value 7!! is 7 times 5 times 3 times 1, which is 105, while 7! is 7 times 6 times 5 times 4 times 3 times 2 times 1, or 5040.
How do I calculate a factorial? ▼
Multiply every whole number from one up to n together. Most calculators provide an exclamation or nPr style key for it.
Why does 0 factorial equal 1? ▼
It is the empty product convention and the value required for combination formulas to work at zero selections.
Common Mistakes to Avoid
- ⚠ Entering values above 170. Factorials grow astronomically fast: 171! exceeds JavaScript’s maximum safe integer, so the calculator caps input at 170. For larger factorials, you need specialized big-number software.’
- ⚠ Forgetting that 0! equals 1 by definition. Entering 0 returns 1, which seems counterintuitive but is essential for combinations, permutations, and series expansions to work correctly.
- ⚠ Confusing n! (all integers multiplied) with double factorial n!! (every other integer). 7! equals 5040, but 7!! equals 105 (7 x 5 x 3 x 1). This calculator computes standard factorial only, not double factorial.
- ⚠ Applying factorial to non integers or negative numbers; the plain definition only covers whole numbers.
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