Permutation or Combination?

Question

How would this problem be solved?

 

Suppose that each of the 24 contestants at the Hunger Games confronts all the other contestants but none die in the process. How many total confrontations would there be for each contestant to confront all the other contestants exactly one time? Use drawings or lists to help organize your thoughts. Show all work.

Answer 1

Ruth,

      If you picture all the contestants on the x and y axis on the integer coordinates then each lattice point is a potential confrontation. Some points should be excluded 1. points on the diagonal represent confrontation with oneself (1,1) (2,2)...

so there are 24 of those. and 2. points say (5,4) and (4,5) should only be counted once. Do you see why?

 

So we have Total confrontations =(24*24-24)/2=(24(24-1))/2=276 which is the same as ₂₄C₂ which is the combination formula.

 

Hope this helps

 

Jim