Formula for the total number of head-to-head matchups in an n candidate election

We can use the 'table' idea to derive the formula for the total number of head-to-head matchups involving n candidates.

Since this table has N rows and N columns, there are N² empty cells below.

But, there are N cells on the main diagonal (candidate 1 vs. candidate 1, candidate 2 vs. candidate 2, etc.) which should be eliminated since they do not represent real matchups. If we take away those N cells (from the original N² cells, we are left with N² -N cells (see the table below).

However, half of the remaining cells are just duplicates of each other (candidate 1 vs. candidate 2 and candidate 2 vs. candidate 1, etc.). So dividing the N² -N cells (from above) by two, we get (N² -N)/2 cells. If we factor out an N, we get an even better way to write this formula: N(N -1)/2 .

Thus, there are N(N -1)/2 total head-to-head matchups in an N candidate election.