



 
Hi Liz, I am going to try a smaller version of the problem.
So I have 5 players named A through E and I need to order the three most popular. I'm going to do this using a tree diagram. My tree starts at the root and has a trunk for each choice of the top ranked player. At the end of each trunk I write the name of the chosen player. No matter which trunk I choose to climb, when I reach the top I have 4 players to choose from for the second ranked player. I construct a branching at the top of each trunk for the four choices. (In my diagram I only drew 2 of the 5 branching at the second level.) You can see that there are 5 × 4 = 20 paths from the root to the top of the second level in the tree. Thus there are 20 ways to produce a ranking of first and second. Again, no matter which path you have chosen to the top of the second level you have 3 choices for the player to rank third. At each position construct three branches to represent the choices. (This time I drew three of the 20 branchings at the top of the second level. Now you can see that there are 4 × 3 × 2 = 60 paths from the root to the top of the third level and thus 60 ways to produce a ranking of first, second and third. Can you see how to extend this procedure to your problem? Penny  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 