Six different books are going on a shelf, and you want to know how many arrangements are possible. The first slot has 6 choices, the next has 5, then 4, then 3, then 2, then 1. Multiply them: 6 x 5 x 4 x 3 x 2 x 1. That product has a name, the factorial, written 6!, and it shows up everywhere counting gets serious. This tool computes factorials instantly, including the large ones where hand multiplication becomes a chore.
The formula
A factorial multiplies every positive whole number from 1 up to n. It grows astonishingly fast, which is exactly why a calculator beats pencil and paper here.
Formula: n! = n x (n - 1) x ... x 2 x 1, with the special case 0! = 1
The 0! = 1 rule looks odd but keeps the math consistent: there is exactly one way to arrange nothing. Each step down multiplies by a smaller number, so 5! = 5 x 4 x 3 x 2 x 1 = 120, and 10! is already 3,628,800. That explosive growth is why factorials dominate counting problems.
How to use it
- Enter n, a whole number (zero is allowed).
- Submit it.
- Read the factorial, shown with the multiplication steps on request.
Factorials are the engine inside bigger counting tools: the permutation and combination calculator uses them directly for nPr and nCr, and the combination calculator leans on them for team and lottery counts.
Worked example
Compute 6! by hand to see the pattern. Write the product: 6 x 5 x 4 x 3 x 2 x 1. Multiply left to right: 6 x 5 = 30. Then 30 x 4 = 120. Then 120 x 3 = 360. Then 360 x 2 = 720. Then 720 x 1 = 720. So 6! = 720, meaning those six books can be shelved in 720 different orders. Now try 0!: by definition it equals 1, the single empty arrangement. A quick check of growth: 7! = 7 x 720 = 5,040, already seven times bigger from a single step up.
Where factorials appear
Every arrangement, ordering, and ranking problem starts here: race finishes, seating plans, password permutations, and shuffles. Probability formulas for card games and lotteries are built from them, and they appear in series expansions in calculus and in the Poisson and binomial distributions in statistics. The factor calculator handles the related but different job of listing divisors, so reach for it when the question is about dividing rather than arranging.
Mistakes to avoid
- Thinking 0! is 0. It is 1, by definition and by consistency with the rest of combinatorics.
- Adding instead of multiplying. 5! is 5 x 4 x 3 x 2 x 1 = 120, not 5 + 4 + 3 + 2 + 1 = 15.
- Canceling carelessly in fractions. In 6! / 4!, cancel 4 x 3 x 2 x 1 from both to get 6 x 5 = 30. Do not divide 720 by 4.
Practical limits
Factorials outgrow ordinary display quickly: 20! has 19 digits, and 171! overflows standard double-precision arithmetic. The tool reports very large results in scientific form. For counting problems with huge n, that compact form is usually what you want anyway, since the exact digit string would fill a page.