Factorials

$n! = n \times (n-1) \times \cdots \times 2 \times 1$ counts the arrangements of $n$ distinct items in a row. It’s the atom of this topic: ${}^nP_r$ and $\binom{n}{r}$ are built from it, and “arrange all of them” questions use it raw.

5 different books on a shelf: $5! = \mathbf{120}$ arrangements.

Two conventions to own: $0! = 1$ (there’s exactly one way to arrange nothing, and the formulas break without it), and factorials grow ferociously ($10!$ is already 3.6 million), which is why exam answers often stay in factorial form unless a number is demanded.

Simplifying factorial fractions: cancel, never expand

The non-calculator paper expects fluent cancellation:

$\dfrac{9!}{7!} = 9 \times 8 = \mathbf{72}$, because $9! = 9 \times 8 \times 7!$, and the $7!$ cancels. $\dfrac{n!}{(n-2)!} = \mathbf{n(n-1)}$, the algebraic version, which appears in equation form: “Solve $\dfrac{n!}{(n-2)!} = 30$” → $n(n-1) = 30$ → $n^2 - n - 30 = 0$ → $n = 6$ (reject $n = -5$: $n$ must be a positive integer, the written rejection is a mark).

Expanding $9!$ fully to divide by $7!$ is the slow, error-prone road; peeling the larger factorial down to the smaller is the entire technique.

Arrangements with structure: blocks and anchors

Factorials combine with the counting principle for structured arrangements:

• Together (the block method): 5 people, two insisting on adjacency → glue them: 4 units arrange in $4!$, the pair arranges internally in $2!$ → $4! \times 2! = \mathbf{48}$
• Apart: total minus together → $5! - 48 = \mathbf{72}$
• Fixed positions: “A must sit at the left end” → anchor A, arrange the rest: $4! = 24$

The block method’s internal factor ($\times 2!$, or $\times k!$ for a $k$-block) is the most-forgotten multiplier in the topic.

Common mistakes

• Factorial fractions expanded instead of cancelled
• $0!$ treated as $0$
• The block’s internal arrangements dropped
• Negative/fractional “solutions” of factorial equations not rejected in writing
• $n!$ confused with $n \times n$ (yes, really, under exam pressure)

Full topic context: P&C notes · next: permutations ${}^nP_r$.