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Fundamental Principle of Counting

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Fundamental Principle of Counting is the basic principle that helps us to count large numbers in a non-tedious way. Suppose we have to guess the pin of a three-digit code so the number of ways we can guess is 1000 this can be seen as the pin can be, 000, 001, 002, ….., 999 any number between 000, and 999 including both numbers. Thus, there are a total of 1000 pin combinations. But using the fundamental principle of counting we can find the total number of possible combinations of the pins in a much simpler way, i.e. for the first place we have 10 different choices from 0-9, similarly for the second place we have 10 choices, and for the third place we have 10 choices. Now the total choices for the pin are 10 × 10 × 10 = 1000 which is much simpler to calculate than to count all possible combinations.

So the fundamental principle of counting helps us to solve various problems of permutation and combination and helps us to make informed choices from all the available choices. In this article, we will learn about the fundamental principle of counting, multiplication rules, addition rules, and others in detail.

Fundamental Principle of Counting

Fundamental Principle of Counting is very helpful in finding all the possible combinations in a particular situation. To understand this principle let’s consider an example. Say a person has 3 pairs of pants and 2 shirts and a question pops up, how many different ways are there in which can he dress? There are three different ways of choosing pants as there are three types of pants available. Similarly, there are two ways of choosing shirts.

Let’s see all the different ways of dressing through a diagram. Considering P1, P2, and P3 as pants and S1, S2 as shirts. The tree is given below lists the range of possibilities

Block Diagram of Fundamental Principle of Counting

Tree Diagram depicting the number of possibilities for the above example 

As shown in the figure, with each type of pant. There are two possible shirts that can be worn. So in total, there are 6 ways of dressing up.

In the problem stated above, the fundamental principle of counting is used to get the result. Here we have used the multiplication rule that states that if an event A can occur in x different ways and another event B can occur in y different ways, then both events can occur simultaneously in x × y different ways. We will learn further about the Multiplication rule further in the article. To learn more about the use of the fundamental principle of counting check the example discussed below,

Example: Count the number of possibilities when a coin is tossed 3 times.

Solution: 

A coin toss can have two outcomes, either Heads(H) or Tails(T), and in case of tossing three coins simultaneously the total number of ways in which this can happen is,

= Total Outcome of First Coin Toss × Total Outcome of Second Coin Toss × Total Outcome of Third Coin Toss

= 2 × 2 × 2

= 8

Thus, tossing three coins simultaneously can have 8 different possible outcomes.

Thus, understanding the above example we can say that the Fundamental Principle of Counting is stated as,

If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrences of the events in the given order is m×n.

This principle can be extended to any finite number of events in the same way. So, for three events this principle becomes, 

“If an event can occur in m different ways, following which another event can occur in n different ways, following both of these events another event happens which can occur in p different ways. So, then the total number of occurrences of the events in the given order is “m × n × p.” 

The fundamental principle of counting is studied under two headings that include,

  • Addition Rule
  • Multiplication Rule

Now learn about these two rules in detail.

Addition Rule

Addition Rule states that for two possible events A and B where A and B both are mutually exclusive events, i.e. they have no outcome in common and if event E is defined as occurring in either event A or event B then the possible number of ways in which event E can occur is,

n(E) = n(A) + n(B)

Where n(A), n(B), and n(E) are the number of events of A, B, and E respectively.

Multiplication Rule

Multiplication Rule states that for “n” mutually independent events, P1, P2, P3, …Pn. The number of in which these events can occur is n(P1), n(P2), n(P3),… n(Pn) respectively. Now we define an event E such that it is happening all the events simultaneously then the number of ways this can happen is,

n(E) = n(P1) × n(P2) …….. × n(Pn)

This is called the multiplication rule of the Fundamental Principle of Counting.

Read More,

Fundamental Principle Of Counting Examples

Example 1: Find the number of four-letter words with or without meaning, which can be made out of letters of the word ROSE, where the repetition of letters is not allowed. 

Solution: 

Number of words that can be formed from these four-letter words is equal number ways in which we can fill __ __ __ __ with letters R, O, S, E. Note that repetition is not allowed. The first place can be filled with any of the four letters, after that second place can only be filled by three letters because we have already used one and repetition is not allowed. Third place can only be filled by two letters and last place will be filled with the last remaining letter. 

So, number of ways in which we can do this are. 4 × 3 × 2 × 1 = 24. 

Note: If the repetition of the letters was allowed we could have always used four letters to fill each place. So 4 × 4 × 4 × 4 = 256. 

Example 2: Given 6 flags of different colors, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?

Solution: 

A signal can be seen like this. 

Arrangement of Flags in Example 2

Here in each position we can use the different colors of flag we are given. So, in the first position we have 6 different choices to make to fill in the position of flag 1. So, in the second position we will have 5 positions to fill because we have already used one color. 

So, total number of ways to fill = 6 × 5 = 30. 

Example 3: How many 2-digit even numbers can be formed from the digits 1, 2, 3, 4, and 5 if the digits can be repeated?

Solution:

 There are five possibilities for putting numbers in each place since the numbers can be repeated. But a constraint is given in the questions which says that the number should be even.

So, all the even numbers have an even digit as the last digit. In the given numbers, only 2 and 4 are two even numbers. So, at the unit’s place in the number, there are only two possibilities while their 5 possibilities for the tens place. 

So, Total number of possible even numbers = 5 × 2 = 10

Example 4: How many positive divisors do 1000 = 2353 have?

Solution:

The positive divisor of 1000 will be in form 2a5b.

Where, a and b will satisfy 0 ≤ a ≤ 3 and 0 ≤ b ≤ 3

It is clear that there are 4 possibilities of a and 4 possibilities of b.

Hence, there are 4 × 4 = 16 Positive Integers of 1000.

Practice Questions on Fundamental Principle of Counting

Q1: What is the possible number of sample space when two die are rolled together?

Q2: In how many ways a 4 digit pin can be set up without repetition of digits.

Q3: In a Felicitation Ceremony in how many ways two guests can shake hand with each other and top three rank holders?

Q4: In how many many ways a six digit PIN Code can be created?

FAQs on Fundamental Principle Of Counting

1. What is Fundamental Principle of Counting?

Fundamental Principle of Counting states that, “If an event can occur in m different ways, and another event can occur in n different ways, then the total number of occurrences of the events is m × n.”

2. What is Use of Fundamental Principle of Counting (FPC)?

Fundamental Counting Principle (FPC) is used to find out the number of possible outcomes for a given situation.

3. What is Fundamental Principle of Counting Formulas?

The fundamental principle of the counting formula is, n(E) = n(P1) × n(P2) …….. × n(Pn)

Where P1, P2,…, Pn are any mutually independent events and n(P1), n(P2), …….. n(Pn) is the number of outcomes of the Events.

4. What are Basic Concepts for Counting?

There are two basic concepts of counting

  • Addition Principle
  • Multiplication Principle


Last Updated : 25 Aug, 2023
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