Factorials — A/L Combined Mathematics

The factorial of a non-negative integer n, denoted by n!, represents the product of all positive integers less than or equal to n: n! = n × (n - 1) × (n - 2) × ... × 2 × 1 for n ∈ ℤ⁺. By axiomatic definition, 0! = 1, ensuring consistency in permutation, combination, and binomial formulas. Factorials satisfy the fundamental recurrence relation: n! = n × (n - 1)!, which allows algebraic simplification of rational factorial expressions without computing massive numbers: for instance, n! / (n - 2)! = n(n - 1) = n² - n.

Check before you answer (A/L Pure Maths Paper 1)

1. Rigorous 0! = 1 Justification: Applying 0! = 1 consistently in combination definitions ⁿC₀ = n! / (0! n!) = 1 and ⁿC_n = 1.
2. Factorial Ratio Simplification: Reducing expressions like (n + 1)! / (n - 1)! directly to (n + 1)n without expanding long products.
3. Factorial Equation Solving: Solving algebraic equations involving factorials (e.g., (n + 2)! = 30 n!) by reducing to standard quadratics.