Factorial Arithmetic — From Intuition to Precision
Imagine you have 3 different books you want to arrange on a shelf. How many different ways can you line them up? You could try listing them: Book A, B, C — or A, C, B — or B, A, C — and so on. If you actually count, you'll find 6 arrangements.
Where does that 6 come from? For the first position, you have 3 choices. Once you pick one, you have 2 choices left for the second position. Then only 1 choice remains for the last spot. So the total is 3×2×1=6.
That product — multiplying a whole number by every positive integer smaller than it, all the way down to 1 — is called a factorial. It's one of the most useful shortcuts in counting.
n!=n×(n−1)×(n−2)×⋯×2×1
The symbol is an exclamation mark: n! is read as "n factorial". It only makes sense for non-negative integers.
The first few values
| n | n! | Why it matters |
|---|
| 0 | 1 | Special case (explained below) |
| 1 | 1 | Only one way to arrange one thing |
| 2 | 2 | Two ways: AB or BA |
| 3 | 6 | Three books, six arrangements |
| 4 | 24 | Four items, 24 arrangements |
| 5 | 120 | Grows fast — five books, 120 ways |
0!=1 is not a guess — it's defined to make formulas work. There is exactly one way to arrange zero objects: do nothing. Also, many formulas like n!=n×(n−1)! would break at n=1 if 0! weren't 1.
The recursive nature
Factorials have a beautiful pattern: every factorial is the current number times the previous factorial.
5!=5×4!
4!=4×3!
3!=3×2!
2!=2×1!
1!=1×0!=1×1=1
This recursive definition is often how you'll compute factorials in problems: n!=n×(n−1)!, with the base case 0!=1.
Why factorials explode so fast
Notice how quickly the numbers grow: 5!=120, 6!=720, 7!=5040, 10!=3,628,800. By 20!, you're already at 2.4 quintillion. This rapid growth is why factorials appear in probability (counting arrangements of decks of cards), combinatorics (choosing teams), and even in advanced mathematics like Taylor series.
A common mistake: thinking n! means n multiplied by something else, like n times some number. It's not — it's the product of all integers from n down to 1. Also, factorials are not defined for negative numbers or fractions in basic arithmetic.
The core idea in one sentence …