Fundamental Counting Principle

Contributor: Mason Smith. Lesson ID: 11311

Tree diagrams are useful, but drawing one can LEAVE you weary! Learn an easier way to figure out how many different combinations of clothes you can wear to the party with a simple counting principle!

categories

High School, Middle School

subject
Math
learning style
Visual
personality style
Lion
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

I would tell you to wait in the queue, but the order doesn't matter.

So far in this Probability: An Overview series of Related Lessons, found in the right-hand sidebar, you have studied experimental and theoretical probability as well as dependency.

Sometimes, when we are talking about the idea of probability, it is possible to create a tree diagram that has all possible outcomes.

However, there are usually way too many possibilities, making a tree diagram impractical. The idea of combinations and permutations offers an opportunity to skip the tree diagram and still get the correct answer without spending hours drawing lines and making sure you haven't skipped an outcome.

Before we can discuss combinations and permutations, we must first understand the idea of the Fundamental Counting Principle (FCP), that states:

  • If there are m ways to choose a first item, and n ways to choose a second item, after the first item has been chosen, then there are m*n ways to choose both items.

However, this can be applied to an infinite number of choices or possibilities, which opens the door for some neat shortcuts.

Look at the tree diagram for this problem:

A new restaurant has opened, and they offer lunch combos for $5.00. With the combo meal, you get 1 sandwich, 1 side, and 1 drink. The choices are listed below:

  • Sandwiches: Chicken Salad, Turkey, Grilled Cheese
  • Chips, French Fries, Fruit Cup
  • Drinks: Soda, Water

Draw a tree diagram to find the total number of possible outcomes.

Solution: There are 18 total combinations.

combo meal tree chart

Instead of drawing out the 18 different possibilities in a tree diagram, we can use the idea of the Fundamental Counting Principle to solve for the number of possible combinations. Try to figure out the answer with a parent or teacher:

  • How many sandwich choices are there? 3
  • How many sides choices are there? 3
  • How many drink choices are there? 2

Using the Fundamental Counting Principle, we can rewrite the number of options for each type of food as a product to find the total number of combinations:

3(sandwiches) * 3(sides) * 2(drinks) = 18 possible combinations

This gives us the same result with much less work and drawing!

Let's do another example:

  • You are going through your suitcase to see how many outfits were packed, and find three pairs of pants, six shirts, and two sets of shoes, since we were packing very light. For now, don't worry about matching. Discuss with a parent or teacher how many different outfits you have.

3 pants * 6 shirts * 2 shoes = 3*6*2 = 36 possible outfits

Now that you have a basic understanding, move on to the Got It? section to check your skills.

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