Fundamental Counting Principle Lesson: Definition, Formula, and Examples

Lesson Overview

• What Is the Fundamental Counting Principle?
• Fundamental Principle Of Counting Examples

The Fundamental Counting Principle simplifies complex counting problems by breaking them down into smaller, more manageable steps. 

This principle is a part of combinatorics, a branch of mathematics that deals with counting, arrangement, and combination. 

What Is the Fundamental Counting Principle?

The Fundamental Counting Principle is a rule that helps us determine the total number of possible outcomes in a situation where there's a need to make a series of choices.  If there are 'm' ways to make one choice and 'n' ways to make another choice after that, then there are m x n ways to make both choices. 

For example - 

Imagine making a sandwich. There are 3 choices of bread (white, wheat, rye) and 2 choices of filling (turkey, ham).

To find the total number of possible sandwiches, multiply the number of bread choices by the number of filling choices: 3 x 2 = 6. So, there are 6 different sandwiches that can be made.

The Fundamental Counting Principle can be extended to more than two choices. If there are also 2 choices of cheese, then the total number of possible sandwiches becomes 3 x 2 x 2 = 12.

Fundamental Counting Rule

The fundamental counting rule consists of two main rules - addition rule and multiplication rule.

Addition Rule

The Addition Rule applies when choices are mutually exclusive (cannot happen at the same time). If there are 'm' ways for one event and 'n' ways for another, there are m + n ways for either to happen.

• Formula: Total outcomes = Outcomes of Event 1 + Outcomes of Event 2 - Outcomes of both events happening together
  

Example:

• Rolling a die to get an even number or a 5: 3 even numbers + 1 outcome for 5 = 4 possible outcomes.

Multiplication Rule

The Multiplication Rule applies when choices are dependent (one affects the other). If there are 'm' ways for one event and 'n' ways for another after the first happens, there are m x n ways for both to happen.

• Formula: P(A and B) = P(A) * P(B|A) (Probability of A and B happening equals the probability of A times the probability of B happening given that A has already happened)
  

Example:

• Drawing two cards from a deck without replacement: 52 choices for the first card x 51 choices for the second = 2652 possible outcomes.

Fundamental Principle Of Counting Examples

Example 1: Lucy has 4 shirts and 3 pairs of pants. How many different outfits can she create?

Solution:

• Lucy has two choices to make: choosing a shirt and choosing pants.
• She has 4 options for shirts and 3 options for pants.
• Multiply the number of options for each choice: 4 shirts x 3 pants = 12
• There are 12 different outfits possible.
  

Example 2: A restaurant offers 5 appetizers, 7 main courses, and 3 desserts. How many different 3-course meals are possible?

Solution:

• There are three choices: appetizer, main course, and dessert - 5 appetizers, 7 main courses, and 3 desserts.
• Multiply the number of options for each choice: 5 x 7 x 3 = 105
• There are 105 different 3-course meals possible.
  

Example 3: A license plate has 3 letters followed by 4 digits. How many different license plates are possible?

Solution:

• 7 choices (3 letters and 4 digits).
• 26 options for each letter (A-Z) and 10 options for each digit (0-9).
• 26 x 26 x 26 x 10 x 10 x 10 x 10 = 175,760,000
• There are 175,760,000 different license plates possible.
  

Example 4: There are 3 different routes to get from city A to city B, and 2 routes from city B to city C. How many different ways can you travel from city A to city C?

Solution:

• There are two choices - A to B, then B to C - 3 routes for the first choice, 2 routes for the second.
• 3 x 2 = 6
• There are 6 different ways to travel from city A to city C.
  

Example 5: If you flip a coin 3 times, how many different sequences of heads and tails are possible?

Solution:

• 3 separate coin flips (each a choice).
• 2 outcomes (heads or tails) for each flip.
• 2 x 2 x 2 = 8
• There are 8 different sequences possible.
  

Example 6: You roll two dice. How many different combinations of numbers can you get?

Solution:

• Each die has 6 possible outcomes.
• 6 x 6 = 36
• There are 36 different combinations.
  

Example 7: You have 5 different books. How many ways can you arrange them on a shelf?

Solution:

• There are 5 options for the first position, then 4 for the second, and so on.
• 5 x 4 x 3 x 2 x 1 = 120
• There are 120 different ways to arrange the books.