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.

If you’ve watched Squid Game on Netflix, you’ll recognize the counting rule in the glass stepping stones scene. Watch this video to see how they used the rule (and how the mathematician in the show got it wrong!):

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## Fundamental Counting Principle Examples

Watch the video for five worked examples of using the counting rule formula:

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### 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

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**Stephanie Glen**. "Fundamental Counting Principle (The Multiplication Counting Rule)" From

**StatisticsHowTo.com**: Elementary Statistics for the rest of us! https://www.statisticshowto.com/fundamental-counting-principle/

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