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The Fundamental Counting Principle

The Fundamental Counting Principle is a method that determines the total number of outcomes of an experiment. The principle states that if there are ways for doing something and ways of doing something else after that, then there are a*b ways of doing both these actions. 


Example:  A quiz consists of five questions with "yes" or "no" answers. In how many different ways can one complete the quiz?
The quiz consists of Question 1, Question 2, Question 3, Question 4, and Question 5. Each question has two possible answers, yes or no. Therefore, the number of different ways that one can do the quiz is:


Example: In how many different ways you can roll a die and toss a coin?
Rolling a die has six outcomes and tossing a coin has two outcomes. Therefore, one can roll a die and toss a coin is:


Example: A bagel shop offers 8 different types of bagels and 15 different types of cream cheese. How many possible combinations of bagel and cream cheese are possible?


Example: Anna plays on her school basketball team. The basketball uniform has:
  • three different sweaters: black, red, and blue
  • three different shorts: black, red, and blue.
How many different variations of the basketball uniform can the coach choose from for each game?


We can use the tree diagram to solve this problem:


We can count the number of branches of the tree diagram which gives us nine as the answer. Or, we can use the fundamental counting principle: there are three choices for sweaters and three choices for shorts which gives a total of nine choices. 
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