11.2 Permutation/Combination Formulas

Subject: Additional Mathematics (4037) | Level: O Level

Part 1: Factorial Notation (n!)

What is a Factorial?

When you multiply a whole number by every whole number smaller than it, all the way down to 1, the result is called a factorial.

The symbol for factorial is ! (an exclamation mark written after the number).

Examples:

The Recursive Rule

There is a very useful shortcut: you can always write a factorial in terms of the one before it.

This means, for example:

• 5! = 5 × 4!
• 7! = 7 × 6!
• 10! = 10 × 9!

This is extremely helpful when simplifying fractions involving factorials, because common factors cancel out.

Example — Simplifying a Factorial Fraction:

Find the value of 8! ÷ 5!

$\frac{8!}{5!} = \frac{8 \times 7 \times 6 \times \cancel{5 \times 4 \times 3 \times 2 \times 1}}{\cancel{5 \times 4 \times 3 \times 2 \times 1}} = 8 \times 7 \times 6 = \mathbf{336}$

The 5! on top and bottom cancel, leaving only the extra factors on top.

Example — Simplifying with Two Factorials on the Bottom:

Find the value of 11! ÷ (8! × 3!)

$\frac{11!}{8! \times 3!} = \frac{11 \times 10 \times 9 \times \cancel{8!}}{\cancel{8!} \times (3 \times 2 \times 1)} = \frac{11 \times 10 \times 9}{6} = \frac{990}{6} = \mathbf{165}$