Fundamental principle of counting

Fundamental Principle of Counting

Fundamental principle of counting :

In this section we shall discuss two fundamental principles.

(i) Principle of addition

(ii) Principle of multiplication.

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

Fundamental Principle of Addition :

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.

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.

To understand this definition, let us consider the following situation For example, in a class there are 2 boys and 3 girls. The teacher wants to select a boy and a girl in a class function. In how many ways can the teacher makes this selection?

Name of boy in the class

Name of girls in the class

Kevin

Karim

Keira

Angola

Cristina

She can select a girl or a boy in the following ways

(Kevin, July), (Kevin, Angola), (Kevin, Cristina), (Karim, July), (Karim, Angola), (Karim, Cristina)

She has 6 ways to make this selection. Since there are 2 boys and 3 girls we can make this set which is containing a boy and a girl easily. For example, if there are 15 boys and 30 girls, this process will be more difficult. To avoid this kind of difficulties we are going to apply the topic fundamental principle of multiplication.

How to decide in which type of question we have to apply the multiplication rule ?

First, we have to analyze whether we have to do both jobs or we can just do only one job. If we have to do both jobs we have to apply this multiplication rule.

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

Here each boy has a chance of getting selected.

(Kevin, Karim, July, Angola, Cristina)

Fundamental principles of counting - Examples

Example 1 :

In a class there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent a competition. In how many ways can the teacher make this selection?

Solution :

Number of ways of selecting a boy = 27

Number of ways of selecting a girl = 14

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 = 27 + 14

= 41 ways

Hence the teacher can make this selection is 41 ways.

Let us look into the next example on "Fundamental principle of counting"

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

Let us look into the next example on "Fundamental principle of counting"

Example 3 :

Given 7 flags of different colors, how many different signals can be generated if a signal requires the use of two flags, one below the other?

Solution :

We have to choose two flags,

Number of ways of selecting 1st flag = 7

After selecting the first flag, we cannot choose the same color flag again.

Number of ways of selecting 2nd flag = 6

Since we have to choose two flags, we have to multiply 7 and 6.

= 7 x 6 = 42

Hence the number of ways of selecting two flags is 42.

Let us look into the next example on "Fundamental principle of counting"

Example 4 :

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

Let us look into the next example on "Fundamental principle of counting"

Example 5 :

A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties and 5 pencil varieties, in how many ways can he select these articles?

Solution :

A person need to buy fountain pen, one ball pen and one pencil. That is we have to do all the works

Number of ways selecting fountain pen = 10

Number of ways selecting ball pen = 12

Number of ways selecting pencil = 5

Total number of selecting all these = 10 x 12 x 5

= 600

Let us look into the next example on "Fundamental principle of counting"

Question 6 :

How many five-digit number license plates can be made if

(i) first digit cannot be zero and the repetition of digits is not allowed.

(ii) the first digit cannot be zero, but the repetition of digits is allowed?