Factorial (Definition)

Statement

The factorial of a non-negative integer \(n\), written \(n!\), is the product of all positive integers from 1 up to \(n\). By definition:

\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \]

Additionally, by special definition:

\[ 0! = 1 \]

Why it works

• Factorials count the number of ways to arrange or order objects.
• For example, \(3! = 3 \times 2 \times 1 = 6\) represents the 6 possible arrangements of 3 objects (ABC, ACB, BAC, BCA, CAB, CBA).
• They also appear in probability, algebra, and series expansions (e.g. binomial theorem, Taylor series).

Recipe (Steps to Apply)

1. Identify the value of \(n\).
2. Multiply all integers from \(n\) down to 1.
3. By definition, \(0! = 1\).

Spotting it

Factorials appear in questions about permutations, combinations, binomial coefficients, probability, and counting arrangements.

Common pairings

• Permutations: \(^nP_r = \frac{n!}{(n-r)!}\)
• Combinations: \(^nC_r = \frac{n!}{r!(n-r)!}\)
• Binomial expansion coefficients: \(\binom{n}{r}\)

Mini examples

1. \(4! = 4 \times 3 \times 2 \times 1 = 24\).
2. \(0! = 1\).

Pitfalls

• Forgetting that \(0! = 1\), not 0.
• Stopping too early when expanding factorials.
• Trying to apply factorials to negative integers (not defined in basic GCSE/secondary maths).

Exam Strategy

• Write factorials as products when first learning to avoid mistakes.
• Simplify factorial expressions by cancelling terms when possible.
• Remember that factorials grow very quickly (e.g. \(10! = 3,628,800\)).

Worked Micro Example

Find \(5!\).

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Summary

The factorial function is the product of consecutive descending integers starting from \(n\). It provides a foundation for permutations, combinations, and many advanced areas of mathematics. Always remember the special case \(0! = 1\).