Factorial Notation

Many of the counting formulas developed later in this chapter involve repeated products such as $n(n-1)(n-2)\cdots 2 \cdot 1$. Because this pattern occurs so often, it is given its own compact notation.

So $1! = 1$, $2! = 2 \times 1 = 2$, $3! = 3 \times 2 \times 1 = 6$, $4! = 4 \times 3 \times 2 \times 1 = 24$, and $5! = 120$. Factorials grow extremely quickly: $10! = 3{,}628{,}800$, and $15!$ already exceeds a trillion — this rapid growth is exactly why counting formulas are left in factorial form rather than expanded into a single huge number wherever possible.

The recurrence relation

Every factorial can be written in terms of the one before it:

This relation is useful both for computing factorials quickly (build up from $1! = 1$) and, just as importantly, for simplifying ratios of factorials without multiplying out either one, e.g.

Cancelling the common factorial tail like this — rather than computing $10!$ and $8!$ separately and dividing — is the standard technique used throughout the rest of the chapter, especially when simplifying the formulas for $^{n}P_{r}$ and $^{n}C_{r}$.

Why $0! = 1$ …