What Is a Factorial, and Why Does It Grow So Fast?
A factorial, written with an exclamation mark, multiplies every whole number from 1 up to n. So 5! is 5 times 4 times 3 times 2 times 1, which is 120. The Factorial Calculator handles any non-negative integer.
It outgrows everything
Factorials grow faster than any exponential, and the speed is hard to believe until you see it.
5! is 120. 10! is 3,628,800. 15! is over a trillion. 20! is about 2.4 times ten to the eighteenth - larger than the number of seconds since the universe began. 70! exceeds ten to the hundredth, which is more than the estimated number of atoms in the observable universe.
That is why a pocket calculator overflows around 69!, and why anything involving factorials of large numbers uses approximations rather than exact values.
Why zero factorial is one
This looks like a trick and it is a necessity.
A factorial counts the number of ways to arrange n things. There is exactly one way to arrange nothing - the empty arrangement - so 0! is 1.
It also falls out of the recursion. Since n! equals n times (n-1)!, then 1! equals 1 times 0!. We know 1! is 1, so 0! must be 1. Defining it any other way breaks the pattern and breaks every formula that uses it.
What they actually count
Arrangements. The number of ways to order n distinct things is n!. Five books on a shelf can be arranged 120 ways. Ten people in a queue, 3.6 million ways.
This is why factorials are everywhere in counting problems. The number of ways to choose and order r things from n uses factorials in both the numerator and denominator, which the Permutation and Combination Calculator handles - and it is also why those numbers stay manageable while the factorials in them do not.
A shuffled deck of 52 cards has 52! possible orders, which is about 8 times ten to the 67. The usual observation is that any well-shuffled deck has almost certainly never existed before in that order, and it is true.
Where they turn up besides counting
Probability, throughout, since probability is largely counting. Series expansions - the exponential, sine and cosine functions are all infinite sums with factorials in the denominators, and because factorials grow so fast those terms shrink quickly, which is what makes the sums converge. Statistics, in the binomial and Poisson distributions.
Common questions
Can you take the factorial of a negative number?
Not with the standard definition. The gamma function extends factorials to non-integers, and it is undefined at negative integers.
What about a decimal?
The gamma function gives a value - and the standard factorial does not.
Why does my calculator give an error above about 69?
The result exceeds what it can represent. Beyond that, approximations like Stirling's formula are used.
How is it calculated so fast?
For small values, directly. The Scientific Calculator has it on the keypad.
Got a factorial to work out?