Factorials

n! — the number of ways to line n things up.

The idea

The factorial of a positive integer $n$, written $n!$ and read $n$ factorial, is the number of ways to arrange $n$ different objects in a row. Whatever goes first, there are $n$ candidates for it; whichever was taken, $n - 1$ remain for second place, then $n - 2$ for third, and so on down to the last object. Each sequence of choices produces a different row, so

$n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1.$

Each factorial is built from the one below it: $n! = n \cdot (n-1)!$, since arranging $n$ objects amounts to choosing what comes first and then arranging the rest.

Factorials grow quickly — $10!$ already exceeds three million — so they are rarely worth multiplying out. In counting formulas they appear in ratios, where the whole of the smaller factorial cancels against the tail of the larger one and a short product remains. Cancel before you multiply.

Ways to work on it

• Walkthrough. What n! counts, why 0! = 1, and cancelling factorial ratios.
• Practice. Evaluate factorials and factorial ratios.
• Hardest. Solve an equation whose unknown sits inside a factorial ratio.

Not sure where to start? Take the ten-question placement test.