Module 8: Permutations, Combinations, and the Binomial Theorem

 

In question 4 of Try This 2 you were asked to make a conjecture about how you can use multiplication to determine the number of different possible outcomes.


To determine the total number of outfits possible, you will multiply the number of shirts by the number of pants and then multiply by the number of shoes or hats. In question 5 you need this product to be 1000 outfits. One possible solution is to have a choice of 10 shirts, 10 pairs of pants, and 10 pairs of shoes. The total number of outfits possible would be 10 × 10 × 10 = 1000.


For question 3.b. in Try This 2 you used a tree diagram to find the number of different outfits or outcomes. Your tree diagram may have looked similar to the diagram shown here.

 

This is a tree diagram showing four choices for pants. From each pair of pants labelled 1, 2, 3, and 4 are two shirts labelled 1 and 2 and from each shirt are branches for two hats labelled 1 and 2. There are a total of 16 branches.

 

This tree identified 16 different possible outfits from 4 pairs of pants, 2 shirts, and 2 hats. You may have noticed that you can also determine the total number of outfits possible by multiplying the number of pairs of pants by shirts and by hats: 4 × 2 × 2 = 16.

 

The fundamental counting principle tells you that if there are a ways to complete one task and b ways to complete another task, then there are a × b ways to complete the two tasks.



Think back to the example from Discover. For Abdy’s PIN, there were two choices for each space in the code. By using the fundamental counting principle, you can calculate the number of possible PINs.

 

 

2 × 2 × 2 × 2 = 16 possible PINs

 

Did You Know?

This is a photo of two sides of a credit card.

Hemera/Thinkstock

Many credit cards and bank cards are equipped with a check digit. This allows you to apply an algorithm to ensure the number on the card is valid. One such algorithm requires you to double every second digit starting on the right. If, when doubling the number, the result is a two-digit number, add the digits together. Find the sum of all these doubled numbers and add this sum to the sum of the digits not doubled. If the card number is valid, the sum should end in zero.

 



textbook

If you would like to see another example of how a tree diagram and the fundamental counting principle are used to determine the number of outcomes, read through “Example 1: Method 1” on page 517 of the textbook.