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Permutations and Combinations are primarily used for counting. In many situations, it is required to find different ways of arrangements of a number of things. If the number is large, counting manually is not possible. Then, we need techniques of permutations and combinations to simplify the counting process.

Fundamental Principles of Counting

There are two fundamental rules of counting which form the foundation of the concepts to follow:

  • Multiplication Rule/AND Rule
    If a certain activity A can be done in “m” different ways and another activity B can be done in “n” different ways, then the total number of ways of doing both A and B simultaneously or sequentially will be m × n.
    In other words, whenever the word “and” connects doing a number of activities, we multiply the number of ways of doing each activity.
    For example: If there are 3 cinema halls to catch a movie and 5 restaurants to go for dinner after the movie, total number of different ways of watching the movie and having dinner afterwards will become 3 × 5 = 15.
    If C1, C2, C3 are three cinema halls and R1, R2, R3, R4, R5 are 5 restaurants, we can see that for each of the 3 choices of cinema hall, we have 5 choices of restaurants to go for dinner. Hence, total number of possible different ways is 15.
  • Addition Rule/OR Rule
    If, out of the two activities, we are going to any one of them, which means either we should do A or we should do B, but not both. Then, the number of ways of doing either A or B will be m + n.
    In other words, whenever the word “or” connects doing a number of activities, we add the number of ways of doing each activity.
    C1, C2, C3      or      R1, R2, R3, R4, R5
    In the previous example, if we can go either to the movie or for dinner, the number of different ways to catch the movie or having dinner will be 3 + 5 = 8.
    If Harish has 6 pants and 10 shirts, then in how many ways can he go to work with a different combination of clothes?
    He has to wear both, pant and shirt. With every pant he can wear any one of the 10 shirts. Since there are 6 pants in all, the number of different combinations of clothes = 10 × 6 = 60

Factorial Notation

The product of first n natural numbers is called factorial n. It is written as n!

i.e.n! = 1 × 2 × 3 × .........× (n 1) × n


Example 6.2: 5! = 5 × 4 × 3 × 2 × 1


Note: 0! = 1. Factorial is defined only for non-negative integers.

Arrangements and Selections

In simple terms, permutations mean arrangements and combinations mean selections.


Let us understand the difference between permutations and combinations or say arrangements and selections. Consider three letters A, B and C.

The above table clearly distinguishes between arrangements and selections. In selections, we count the type of objects. Hence, (A B) and (B A) are considered as only one selection or combination.


In case of arrangements, the order in which the objects are kept in a row is important. Thus, (A B) and (B A) are two different arrangements.

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