Probability and Statistics > Probability > Fundamental Counting Principle

## Fundamental Counting Principle Definition.

The 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:

- Choose a sandwich.
- Choose a side.
- Choose a dessert.
- 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?**

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## References

Dodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.

Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company