Fundamental Counting Principle (The Multiplication Counting Rule)

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Fundamental Counting Principle Definition.

fundamental counting principleThe Fundamental Counting Principle (also called the counting rule) is a way to figure out the number of outcomes in a probability problem. Basically, you multiply the events together to get the total number of outcomes. The formula is:

If you have an event “a” and another event “b” then all the different outcomes for the events is a * b.


Fundamental Counting Principle Examples

Fundamental counting principle: Example problem #1

A fast-food restaurant has a meal special: $5 for a drink, sandwich, side item and dessert. The choices are:

  • Sandwich: Grilled chicken, All Beef Patty, Vegeburger and Fish Filet.
  • Side: Regular fries, Cheese Fries, Potato Wedges.
  • Dessert: Chocolate Chip Cookie or Apple Pie.
  • Drink: Fanta, Dr. Pepper, Coke, Diet Coke and Sprite.

Q. How many meal combos are possible?
A. There are 4 stages:

  1. Choose a sandwich.
  2. Choose a side.
  3. Choose a dessert.
  4. Choose a drink.

There are 4 different types of sandwich, 3 different types of side, 2 different types of desserts and five different types of drink.

The number of meal combos possible is 4 * 3 * 2 * 5 = 120.

Fundamental counting principle: Example problem #2.

Q. You take a survey with five “yes” or “no” answers. How many different ways could you complete the survey?

A. There are 5 stages: Question 1, question 2, question 3, question 4, and question 5.
There are 2 choices for each question (Yes or No).
So the total number of possible ways to answer is:
2 * 2 * 2 * 2 * 2 = 32.

Example problem #3.

Q: A company puts a code on each different product they sell. The code is made up of 3 numbers and 2 letters. How many different codes are possible?
A. There are 5 stages (number 1, number 2, number 3, letter 1 and letter 2).
There are 10 possible numbers: 0 – 9.
There are 26 possible letters: A – Z.
So we have:
10 * 10 * 10 * 26 * 26 = 676000 possible codes.

Fundamental Counting Principle Problems: Your turn!

Click on the question to reveal the answer.

Question 1: You toss three dimes. How many possible outcomes are there?

Question 2: Your school offers two English classes, three math classes and three history classes. You want to take one of each class. How many different ways are there to organize your schedule?

Question 3: A wedding caterer gives you three choices for the main course, six starter choices and five options for dessert. How many different meals (made up of starter, dinner and dessert) are there?

Question 4: You take a multiple choice test made up of 10 questions. Each question has 4 possible answers. How many different ways are there to answer the test (assuming you don’t leave a question blank)?

Question 5: An online company is offering a date night special: pick one movie from four choices, one restaurant from six choices and either flowers, chocolates or wine. How many possible date night options are there?

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Dodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.
Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company

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