1. What Is a Permutation?
A permutation is an arrangement of objects where the order of the objects is important. Changing the order creates a new arrangement.
We use permutations when the sequence of items matters, like arranging books, forming numbers, or seating people in a row.
1.1. Understanding 'Order Matters'
If swapping two objects changes the outcome, then it is a permutation.
Example: The arrangements ABC and ACB are different because the positions of B and C are swapped.
So, any time the arrangement changes when the order changes, we use permutations.
1.2. Simple Real-Life Examples
Example: Seating 3 friends A, B and C in a row. ABC, BAC, CBA etc. are all different.
Example: Creating a 3-digit number using digits 1, 2 and 3. 123 and 321 are different because the digits are in different positions.
2. Permutations of Distinct Objects
When all objects are different, counting arrangements becomes straightforward using factorials.
2.1. Arranging All n Distinct Objects (n!)
If we have \(n\) distinct objects, then the number of ways to arrange all of them is:
\( n! \)
Example: 4 different books can be arranged in \(4! = 24\) ways.
2.2. Arranging r Out of n Objects (nPr)
If we choose \(r\) objects from \(n\) and arrange them, the number of ways is:
\( ^nP_r = \frac{n!}{(n-r)!} \)
Example: Number of ways to arrange 2 out of 5 students is \( ^5P_2 = 5 \times 4 = 20 \).
2.3. Deriving the Formula Using Counting Principle
To fill the first position we have \(n\) choices. For the second position, \(n-1\) choices remain. For the third, \(n-2\) choices remain, and so on.
So, for \(r\) positions:
\( n \times (n-1) \times (n-2) \times \dots \text{(r factors)} = \frac{n!}{(n-r)!} \)
3. Permutations with Repetition Allowed
Sometimes objects can be used more than once. In such cases, we do not reduce choices after each step.
3.1. Using Digits and Letters with Repetition
Example: Forming a 3-digit number using 0–9 where repetition is allowed.
For each digit we have 10 choices.
Total = \(10 \times 10 \times 10 = 1000\).
3.2. Counting Using the Multiplication Principle
When repetition is allowed:
Number of arrangements of length \(r\) from \(n\) symbols = \(n^r\).
Example: 4 letters A, B, C, D → number of 2-letter repeated codes = \(4^2 = 16\).
4. Permutations with Identical (Repeated) Objects
When some objects are the same, swapping identical objects does not create a new arrangement. So we divide to avoid over-counting.
4.1. Words with Repeated Letters
Example: In the word MOM, M is repeated twice. Total arrangements are:
\( \frac{3!}{2!} = 3 \)
Arrangements: MOM, MMO, OMM.
4.2. General Formula for Repeated Items
If a set has \(n\) objects where some are identical:
\( \frac{n!}{p_1! \cdot p_2! \cdot p_3! \dots} \)
Example: In BALLOON, the letters L appear twice and O appears twice:
\( \frac{7!}{2! \cdot 2!} \)
5. Circular Permutations
In circular arrangements, there is no fixed left or right end, so one arrangement can look like another after rotation.
5.1. Arrangements Around a Circle
The number of ways to arrange \(n\) distinct objects around a circle is:
\( (n-1)! \)
This is because one position is fixed to avoid rotations counting as different.
5.2. When Clockwise and Anticlockwise Are Considered the Same
If mirror images are also considered identical (like a necklace), then the number of arrangements becomes:
\( \frac{(n-1)!}{2} \)
6. Common Permutation Situations
Permutations appear in many everyday counting problems. Here are some common patterns.
6.1. Forming Numbers from Digits
Using digits with or without repetition to form numbers is a very common permutation problem.
Example: Form 3-digit numbers using digits 1, 2, 3, 4 without repetition → \( ^4P_3 = 24 \).
6.2. Seating and Ordering Problems
Arranging people in a row, creating line-ups, or ordering items in a list are all permutation problems.
Example: Seating 5 friends in a row → \(5! = 120\) ways.