Cambridge O Level Additional Mathematics · Syllabus 4037 · Permutations and Combinations
Factorial
What is Factorial?
For a positive integer n, n factorial, written n!, is the product of every positive integer from n down to 1, so that 5! is 5 times 4 times 3 times 2 times 1, which is 120. It counts the number of ways of arranging all n distinct objects in order. The value of 0! is defined to be 1, because there is exactly one arrangement of no objects at all, and this convention is what makes the permutation and combination formulas work at their end points. Factorials satisfy the recursion n! equals n times (n minus 1) factorial, which is what allows a quotient such as 8! divided by 5! to be cancelled down to the short product 8 times 7 times 6.
This definition is part of the Permutations and Combinations chapter in Cambridge O Level Additional Mathematics.
Common mistakes with Factorial
- Evaluating \({}^nP_r\) or \({}^nC_r\) with \(r>n\). Why it fails \((n-r)!\) would be the factorial of a negative integer, which does not exist. It is also plainly impossible: you cannot fill \(5\) slots from \(3\) objects without reusing one, and reuse is forbidden. Fix If a case in a decomposition needs \(r>n\), that case simply cannot occur and contributes \(0\) — drop it, do not force it.