11.3 Arrangement and Selection Problems

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📝 Main Content

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Section 1: Quick Recap — Factorial Notation

Before solving arrangement and selection problems, you need to understand factorial notation.

The factorial of a number n (written as n!) means you multiply that number by every positive whole number below it, all the way down to 1.

$n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$

Examples:

• $3! = 3 \times 2 \times 1 = 6$
• $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
• $0! = 1$ ← This is a special rule you must memorise

A useful shortcut:

$n! = n \times (n-1)!$

For example: $7! = 7 \times 6!$

This shortcut is very helpful when simplifying fractions involving factorials.

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The Permutation Formula

The number of ways to arrange r items chosen from n items (where order matters) is:

$^nP_r = \frac{n!}{(n-r)!}$

• n = total number of items available
• r = how many you are choosing and arranging

Step-by-step logic behind the formula:

Imagine you have 8 letters and you want to arrange 3 of them in a row.

• For the 1st space: you have 8 choices
• For the 2nd space: 7 choices remain
• For the 3rd space: 6 choices remain

Total = $8 \times 7 \times 6 = 336$

Using the formula: $^8P_3 = \dfrac{8!}{(8-3)!} = \dfrac{8!}{5!} = 8 \times 7 \times 6 = 336$ ✓