## Lesson 4.8 – Combinations and Probability

• Compute the number of combinations of n individuals taken k at a time.
• Use combinations to calculate probabilities.
• Use the multiplication counting principle and combinations to calculate probabilities.

Activity: How many ways can you order a hamburger?

Today’s activity is based on a take out bag that Wendy’s had a few years ago.  In fact, Luke actually had it posted on his wall for a while.  The bag says, “We figured out that there are 256 ways to personalize a Wendy’s hamburger. Luckily someone was paying attention in math class.”  So we got to wondering, how many ways can you order a hamburger.  First, we need to recognize that this is NOT a permutation because the order does not matter.  Ketchup and then mustard tastes the same as mustard then ketchup.  So if we don’t care about the order, how do we calculate it?

We worked through one particular hamburger (ketchup, mustard, lettuce) and thought about how many possible ways could we order these toppings.  We listed out all possible orders and then realized that there are 3! ways to order the 3 toppings.  We used this to think about, if there are 9 toppings, how many ways can you choose 3?  You can see our work below.

Because of this thought process, we like to calculate combinations by taking the number of permutations divided by the factorial.  It just makes more sense to students.  There is meaning behind the formula.  We use the phrase, “Divide out the repeats.”

To complete the original question, we made a table to organize all the different combinations of toppings that can be chosen.  You’ll see that the bag was wrong, there are 512 ways to order a hamburger.

We modified today’s activity based off another we found online.  Read through the whole post! There are some cool ideas for extension and handouts.

Notes

When planning for this lesson, think about what level your students are at.  If you think they’ll struggle, you may want to employ a think aloud for #2 and #3.  We encourage you though to let the students grapple with the concepts.  Be conscious of your wait time and try to stretch their thinking.  Lastly, students will notice the repetitive nature of combinations which is a great extension to discuss.