FUNDAMENTAL PRINCIPLES OF COUNTING

In this section we shall discuss two fundamental principles.

(i) Multiplication

(ii) Addition

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

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?

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.

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?

Here each boy has a chance of getting selected.

(Kevin, Karim, July, Angola, Cristina)

Solved Problems

Problem 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.

Problem 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

Problem 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.

Problem 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

Problem 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

Problem 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?

Solution :

Numbers can be filled in the places are 0,1,2,3,4,........9

___ x ____ x ____ x ____ x ____

Number of options we have for first place  =  9 (except 0)

Since repetition is not allowed the second dash is having 9 options (including 0 except the number filled in the first dash).

Like wise the third, fourth and fifth dashes are having 8, 7 and 6 options respectively.

Total number of ways  =  9 x 9 x 8 x 7 x 6

  =  27216

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