Question 9. Anna has 5 circular disks, each of a different size. She decides to build a tower using 3 of her disks so that each disk in her tower is smaller than the disk below it. How many different towers could Anna construct? If we lay down our disks like so and label them A, B, C, D, and E, with A being the largest and C being the smallest, this will make it easier for us to solve this question. Start off with tower A, B, C. C is the smallest in this tower and A is the biggest. Now we can do the same thing replacing C with D, and again replacing D with E. All of these are valid towers as Anna wants. Next we can make a tower with base A, C, and then add D, and then we could also replace D with E. Finally, if we want A as our base, we can make tower A, D, E. These are all the towers we can make with A being on the bottom. Next we can make some towers with B at the bottom. We'll make B, C, D, and then B, C, E. We could also make B, D, E. Finally, we can make one more tower with C being as our base. It will be tower C, D, E. This will give us our answer if we total up the amount of towers, which is 10. So the answer is D.

circular disks

tower construction

different sizes

combinatorics

disk arrangement