Factorial Calculator: Exact Factorials With Digit Counts

A factorial counts arrangements. The factorial of n, written n with an exclamation mark, is the product of every whole number from 1 up to n. So 5 factorial is 5 times 4 times 3 times 2 times 1, which equals 120. That is the number of ways to order 5 distinct items, because there are 5 choices for the first position, 4 for the second, 3 for the third, 2 for the fourth, and 1 for the last.

By definition, 0 factorial equals 1. There is exactly one way to arrange nothing, and that convention makes formulas for combinations and permutations work cleanly at the edges. Factorials grow extremely fast: 10 factorial is already 3,628,800, 20 factorial is about 2.43 times 10 to the power 18, and 100 factorial has 158 digits.

For choosing items without caring about order, a permutation and combination calculator uses factorials directly.

The definition is a simple recurrence: n factorial equals n times the factorial of n minus 1, with 0 factorial equal to 1. So 6 factorial is 6 times 5 factorial, which is 6 times 120, which is 720. That recurrence is also how this calculator builds the answer, multiplying upward from 1 with exact integer arithmetic, so the result has no rounding error.

Trailing zeros come from factors of 10, and each factor of 10 needs one 2 and one 5. Since factors of 2 are plentiful, the count of trailing zeros is the count of factors of 5: floor of n over 5, plus floor of n over 25, plus floor of n over 125, and so on. For 20, that is 4 plus 0, which is 4, and indeed 20 factorial ends in 4 zeros.

Example 1: 5 factorial. Multiply 5 times 4 times 3 times 2 times 1 to get 120. Example 2: 0 factorial. By definition, the answer is 1. Example 3: 10 factorial. The product of 1 through 10 is 3,628,800, which has 7 digits and 2 trailing zeros.

Example 4: 20 factorial. The exact value is 2,432,902,008,176,640,000, with 19 digits and 4 trailing zeros. Example 5: how many ways can 4 books be ordered on a shelf. The answer is 4 factorial, which is 24.

Factorials outgrow exponentials, which is why they appear in probability problems with surprisingly large answers. The number of ways to shuffle a deck of 52 cards is 52 factorial, a number with 68 digits, far larger than the number of atoms in the observable universe is often quoted to be, and far beyond anything a human could list.

Stirling's approximation estimates large factorials: n factorial is about the square root of 2 pi n, times n over e, all to the power n. For 20, that estimate is close to the exact value, and for larger n the relative error shrinks, which is why statisticians use it when exact values are unwieldy.

The most common mistake is adding instead of multiplying: 5 factorial is not 5 plus 4 plus 3 plus 2 plus 1, which is 15, but the product, which is 120. Another mistake is starting from 0: multiplying by 0 would give 0, so the product runs from 1 to n, with 0 factorial handled separately as 1.

A third mistake is applying factorials to fractions or negatives. Factorials are defined for whole numbers from 0 upward in this calculator. The gamma function extends the idea to other values in advanced mathematics, but it is not a factorial in the ordinary sense, and it is not what counting problems need.

Factorials count permutations, combinations, and probability outcomes: lottery odds, seating arrangements, password possibilities, and card shuffles all start with a factorial. They also appear in series expansions in calculus, such as the Taylor series for the exponential function, where each term divides by a factorial.

For prime-related counting questions, a prime number calculator checks primality and factors. Use this factorial calculator whenever a problem asks how many orders or arrangements a set of distinct items has.