Factorial Notation

Repeatedly applying the multiplication principle to arrange a set of distinct objects produces the same kind of product again and again — $n \times (n-1) \times (n-2) \times \cdots \times 1$. Because this product shows up so often in counting problems, it is given its own compact notation: the factorial.

Definition

For a positive integer $n$, the factorial of $n$, written $n!$ (read "$n$ factorial"), is defined as

By convention, $0! = 1$. This is not something to derive from the product formula (the product would be empty) — it is a definition adopted purely so that the permutation and combination formulas below stay valid even when $r = 0$ or $r = n$.

Recurrence relation

This relation is often the fastest way to simplify an expression that mixes two different factorials, since it lets us peel off (or build up) factors one at a time instead of expanding both factorials in full.

A short table of values

Notice how quickly factorials grow — this rapid growth is exactly why counting problems need a compact notation rather than writing out every arrangement by hand; a Gujarat Std-11 Commerce student is expected to compute factorial-based expressions confidently, not just quote the definition.

Simplifying factorial expressions …