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. 

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 2^a5^b.

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.

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.