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Counting Rules and Cookout Milkshakes

I am always trying to find problems that my students want to solve.  As a resident of South Carolina we have the lovely fast food restaurant: Cookout.  Cookout is known for cheap food and their fancy milkshakes.  There are 40 different flavors of a fancy milkshake.  Most people assume that there are 40 different milkshakes that you can order, however, to the consumers delight you can combine multiple flavors together to create your shake.  Yes you can order a milkshake with all 40 flavors, rumor has it this is a very bad decision.  This here sets the stage for a conversion about permutations, combinations, and sums as well as a lesson that students remember.

I posed this problem to my students and to sweeten the deal ordered all of the students fancy shakes from Cookout.  Here is the Cookout Menu (compliments of the Cookout website).  All of the fancy milkshake favors are on the rights.As I decided to get each student a milkshake that meant I had to take a quick trip to cookout before class during my planning.  There are a few concerns with ordering 19 milkshakes: How do you carry them?  How do you keep them from melting?  So I walked into Cookout during prime lunch time with a cooler, a list of milkshake orders, and an envelope of $1 and change. Needless to say it was quite an experience, but a successful one thanks to the wonderful staff at cookout.

imageWhile enjoying some delicious milkshakes we began the process of finding how many combinations of milkshakes we could order.  There are 40 different flavors and you can order any number of flavors in your milkshake.  We began our problem by determining how many 1 flavor milkshakes we could order, which was easily determined as 40.  Next we moved to the number of 2 flavor milkshakes we could order.  Since the order of flavors does not matter as the flavors are mixed together we used a combination for 40 choose 2 for a total of 780 possibilities! (This means there is also 780 combinations of 39 flavor milkshakes)  We continued using combinations for each number of flavors: 3,4,5, etc all the way to 40.  As our numbers grew in size we used Excel to compute the numbers and tally the sum.  I will spare you all the calculations and I hope this has you thinking enough to explore this idea on your own and discover the number of cookout milkshakes for yourself.

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