Factorials — A/L Combined Mathematics

What must I remember about Factorials?

• Definition of Factorial: For a non-negative integer n, n factorial (denoted n!) is the product of all positive integers less than or equal to n: n! = n * (n - 1) * (n - 2) * ... * 3 * 2 * 1.
• Zero factorial convention: 0! is defined to be 1 (0! = 1), ensuring algebraic consistency in permutations, combinations, and binomial coefficients.
• Recursive recurrence relation: n! = n * (n - 1)! for all positive integers n ≥ 1.
• Simplification of factorial fractions: Express larger factorials in terms of smaller factorials: n! / (n - r)! = n(n - 1)(n - 2)...(n - r + 1).
• Domain limitation: Factorial function is strictly defined for non-negative integers; negative integers have undefined factorials (poles).

Test yourself (1 minute)

1. Why 0! = 1

   From recurrence relation (n - 1)! = n! / n; substituting n = 1 gives 0! = 1! / 1 = 1.

2. Factorial growth rate

   Factorials grow faster than any exponential function aⁿ; n! >> 2ⁿ for large n.

3. Prime factorization of n! (Legendre's Formula)

   The exponent of prime p in n! is E_p(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ...

Practise Factorials questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 23

What should I study first?

What comes next?

Next in the syllabus

Where does Factorials fit in the syllabus?

Year 2 · Term 1