COUNTING PRINCIPLE

Counting principle :

In this section, we shall discuss two fundamental principles.

(ii) Principle of multiplication.

These two principles will enable us to understand permutations and combinations and form the base for permutations and combinations.

If there are two jobs such that they can be performed independently in "m" and "n" ways respectively, then either of the two jobs can be performed in (m + n) ways.

Let us consider the following example to understand the above concept better.

Example (OR) :

One can go to school by bus where there are 5 buses or  by auto where there are 4 autos.

In the given two jobs "going to school by bus" and "going to school by auto", only one of the jobs can possibly be done and not both.

That is, the student can go to school by bus or by auto.  So, we have to use the concept "OR" and we have to add  the ways to do the first job and second job.

Total no. of ways of going to  school by bus or by auto is

=  5 + 4

=  9 ways

Fundamental Principle of Multiplication :

If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any one of these m ways, second job can be completed in n ways; then the two jobs in succession can be completed in m × n ways.

Let us consider the following example to understand the above concept better.

Example (AND) :

One can go to school by 5 different buses and then come back by 4 different buses.

Here, both the jobs "going to school" and "coming back from school" have to be done. So, we have to use the concept "AND" and we have to multiply the ways to do the first job and second job.

Total no. of ways of going to  and coming back from school is

=  5 x 4

=  20 ways

Let us look at some more examples on the above concepts.

Counting principle - Examples

Example 1 :

In a class there are 20 boys and 10 girls. The teacher wants to select either a boy or a girl to represent the  class in a function. In how many ways can the teacher make this selection?

Solution :

Number of ways of selecting a boy  =  20

Number of ways of selecting a girl  =  10

From the given question, we come to know that we can select a boy or a girl. That is, it is enough to do one of the works.

So, we have to use the concept principle of addition.

Total number of ways to make this selection  =  20 + 10

=  30 ways

Hence the teacher can make this selection is 30 ways.

Example 2 :

A room has 10 doors. In how many ways can a man enter the room through one door and come out through a different door?

Solution :

Here we have two job,

(i) Entering into the room  =  10 ways

(ii) Come out from the room  =  9 ways

A person must do the above two jobs. So, we have to multiply the number of ways of each work.

Hence, the total number of ways to do the work  =  10 x 9

=  90

Example 3 :

How many words (with or without meaning) of three distinct letters of the English alphabets are there?

Solution :

Total number of English alphabets  =  26

Here we have to fill up three places by distinct letters.

____   ____   ____

We can fill up any one of the 26 alphabets.

So, there are 25 ways of filling up the second place.

Now, the second place can be filled up by any of the remaining 25 letters.

After filling up the first two places only 24 letters are left to fill up the third place.

So, the third place can be filled in 24 ways.

Hence the required number of ways  =  26 x 25 x 24

=  15600 After having gone through the stuff given above, we hope that the students would have understood "Counting principle".

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