Pascal’s Triangle and Set Theory

Prepared by Shari Davis and Katherine Smith as part of a series of supplemental critical thinking explorations to promote discovery and deeper comprehension.
This project was supported by a 2017 Faculty Innovator Grant through Old Dominion University’s Center for Learning and Teaching.

Part 1

Getting Ready

Students should have familiarity with basic set theory including the definition of a set, basic properties, and how to form subsets of a given set.

Video

Students should watch the following video: Critical Thinking Exploration 3 Part 1 Video

Follow-up

After watching the video, students should be given time to think about the connection between 11 raised to various powers and the rows of Pascal’s Triangle. This could take place in class, after class or using an online discussion board.

A sample discussion board prompt would be: How are the powers of 11 related to Pascal’s Triangle? Can you find 11⁵? Do you think the pattern continues?

Part 2

Getting Ready

Students should have viewed Part 1 of this exploration and had time to complete the follow-up activity. Additionally for the follow-up activity, students should be familiar with the formula for calculating the number of subsets from a given set.

Video

Students should watch the following video: Critical Thinking Exploration 3 Part 2 Video

Follow-up

After watching the video, students should be given time to think about whether or not the method we just learned using Pascal’s triangle is consistent with the formula for calculating the number of subsets. This could take place in class, after class or using an online discussion board.

A sample discussion board prompt would be: Another way to find the total number of subsets that can be formed from a given set is to evaluate 2ⁿ where n is the number of elements. Does this agree with the method we just learned using Pascal’s Triangle? Check the first few rows to make sure.

Part 3

Getting Ready

Students should have viewed Part 2 of this exploration and had time to complete the follow-up activity.

Video

Students should watch the following video: Critical Thinking Exploration 3 Part 3 Video

Follow-up

After watching the video, students should be given time to practice the questions at the end of the video or additional similar questions. This could take place in class, after class or using an online discussion board.

A sample discussion board prompt would be: How many subsets with 4 elements can be formed from a set with 6 elements? How many subsets with 123 elements can be formed from a set with 123 elements? Hint: For the second question, you should not write out the rows of Pascal’s Triangle to get your answer.

Part 4

Getting Ready

Students should have viewed Part 3 of this exploration and had time to complete the follow-up activity. Additionally, it will be helpful if students are familiar with the definition of n!.

Video

Students should watch the following video: Critical Thinking Exploration 3 Part 4 Video

Follow-up

After watching the video, students should be given time to think about and discuss other examples where they may use combinations. This could take place in class, after class or using an online discussion board.

A sample discussion board prompt would be: Can you think of any other practical examples where you may need to know how many subsets of a particular size you can choose from a set with a given number of elements?