Permutations

Definition of Permutations

The number different ways that a certain number objects can be arranged in order from a larger number of objects. That arranging people, digits, numbers, alphabets, letters and colors.

Example

I had a shelf of 5 different books, and I wanted to know : in how many different orders can I put these 5 books? Another way to say that is : 5 books have how many different permutations? In order to answer this question, we need an odd math symbol : the factorial. It is written as an exclamation sign, and it means : the product of that number and all the positive integers below it, down to 1.

Formula of Permutations

Keywords of Permutations

• Arrangements
• Arrange
• Schedule
• Order

Types of Permutations

• Repetition is allowed : such as the lock above. It could be “333”.
• No Repetition : There are the first three people in a running race, you can’t be first and second.

Permutation with Repetition

These are the easiest to calculate.

Which is easier to write down using an exponent of  :

Example

Formula of Permutations in Repetition

Permutation without Repetition

In this case, we have to reduce the number of available choices each time.

Example

Without repetition our choices get reduced each time.

But how do we write that mathematically? Answer : we use the “factorial function”.

So, when we want to select all of the billiard balls the permutations are :

Formula of Permutations without Repetition

Example

Notation

Instead of writing the whole formula, people use different notations such as these :

Example 1

Example 2

References :

• https://www.mathsisfun.com/combinatorics/combinations-permutations.html
• http://magoosh.com/gmat/2012/gmat-permutations-and-combinations/