## Arrangements

Situations often occur where a number of items have to be selected from a larger set of items. There can be different requirements on how these items are to be selected and arranged and there are a number of techniques and formulae which can help to do this.

### The Multiplication Principle

This states that:

 If there are r events which can take place in order and the first event can occur in n1 ways the second event can occur in n2 ways until the r th event which can occur in nr ways then the events can occur in n1 × n2 × n3 × ....x nr ways.

A couple of simple examples will illustrate this concept:

Example 1 How many three digit numbers can be made from the digits 1 to 6 inclusive if no digit is to be repeated?

The first digit can be filled by any one of 1, 2, 3, 4, 5 or 6
The second digit can be selected from the remaining 5 digits.
The third digit can be chosen from the remaining 4 digits.

By the multiplication principle there are 6 × 5 × 4 = 120 three-digit numbers available.

Example 2 New Zealand number plates (not personalized plates) are made up of two letters followed by four numbers. How many different number plates are possible in theory?

By the multiplication principle there are 26 × 26 × 10 × 10 × 10 × 10 =6, 759, 324 possible plates.

### Factorial

Before we proceed any further it is necessary to define the meaning of n factorial or n!

n factorial is the product of all of the natural or counting numbers less than or equal to n.

e.g. 7 Factorial = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

0! is defined as 1.

On a calculator the factorial button is usually given by x!

Expressions involving factorials can often be simplified by cancelling.

Example ### Some Arrangements

Some of the commonly occurring situations are covered below:

Different objects in a line
 The number of ways of arranging n unlike objects in a line is n!

Example
In how many ways can four different CDs be arranged on a shelf?
There are 4! = 24 ways in which the CDs can be arranged.
Objects in a line if some are alike
 The number of ways of arranging n objects in a line if p of them are the same is Example
How many different four letter "words" can be made from the letters of the word SARA?
The number of words = 4! / 2! = 12
Objects in a line if there are more than one similar types

 The number of ways of arranging n objects in a line if p of them are of one type and q of them are of another ... is Example

In how many different ways can all of the letters of the word STATISTICS be arranged?
The number of ways is Objects in a cirlce or ring
 The number of ways of arranging n unlike objects in a ring is (n − 1)!

Example
In how many ways can 5 people be seated around a circular table.

There are (5 − 1)! ways = 24 ways