Expert: Sherry Wallin - 6/10/2011

Question
Hello again,

I'm in the final week of this abstract algebra class and am stuck again, so I'm hoping you can provide some guidance (you really helped me out a couple of weeks ago). This time, I'm having a hard time with Burnside's Counting Theorem.

We were asked to find a real-life application of the theorem, which was easy enough. My example was that, when I worked in an ice cream shop during high school, the most popular order was a double scoop cone. There were 25 different flavors of ice cream to choose from for your two scoops, so obviously Burnside's theorem can tell us how many distinguishable possibilities there are for my cone. However, I don't quite grasp how to apply the theorem to actually get the results. Would you be able to help me work through this?

Thanks in advance,
Nik

Answer
Hi Nik~

I don't think your example warrants Burnside's Theorem because the answer to your problem is really quite simple. There are 25 choices for the first scoop and also 25 choices for the 2nd scoop (because you did not make a restriction on having a double scoop of one flavor) so there were 25^2 =  625 ways to make a double scoop cone. Now if you were to introduce some other variables you might make this a Burnside's theorem problem. For example what if you were to alphabetize the flavors names and make it a requirement that after the first scoop was chosen only flavors alphabetically further down the list could be chosen? This is similar to if the flavors were in a long line, in some kind of order and you said to a customer once you've chosen flavor x in position y, then you must choose a flavor x further down the line. A concrete of what I am saying is that you have 25 flavors in a line. Suppose a customer chose the 6th flavor, then their second scoop could only be chosen from position 7,8,9,...,25. Think about this and let me know what you  decide to do.

Math Prof