Permutation and Combination

Permutation and combination cannot be defined separately, they are the process by which the objects are selected from the sets without the replacement to form the subset, the cases in which the order of selection is important it is called as permutation.

When the order of selection is not the important factor it is called as the combination.


Mathematical Formula for Permutation:

The permutation for n objects taken r at a time is given by,


Let us understand the permutation in the better way with the below example,
If we have 4 students and we need to select 2 among them for specific roles, the permutation can be given by,
4P2 = 4! / 2!


The combination for n objects taken r at a time is given by,


Below is the simple example to understand the combination in the better way,
If we want to select any 4 balls out of 7 then the combination is given by,
7C4 = 7! / (7-4)! 4! = 35
This denotes that balls can be selected in 35 different ways.

Case 1: Types of Permutation

There are generally two types of permutation,

  1. Repetition is allowed
  2. No Repetition
  3. Permutation of multisets.

No Repetition

For Example: How many 3 letter word can be made from the 5 letter word without repetition is given by
P (n, r) = 5! / (5-3)! = 60
In the cases, were repetitions are not allowed the choices are seemed to get reduced every time.

Consider the same above case but with the condition that the repetition is allowed is given by,
nr = 53= 125

Permutation of Multi Sets
If an operation can be decided to be performed in m ways then the second operation can be performed in n ways, then the number of ways of performing the two operations together is given by m x n.

Case 2: Types of Combination

The combination also found to have 2 types

  1. Repetition is allowed
  2. No Repetition

In the cases where repetition is allowed the best examples are the coins in your pocket, they can be either 5, 5, 5,10 or some other.
In the cases where the repetition are not allowed the easy example to give is the numbers in the lottery ticket. The numbers in the lottery ticket can never be repeated.


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