Counting on Determinants

See Figure 3 for an example. We define an n-path to be a collection of n paths from a set of n origins to a set of n destinations. To compute the number of 4-paths in our example, we first find the number of ways that each ant can reach each morsel. In the example these numbers can be computed easily using calculations similar to those that arise in Pascal’s triangle (see Figure 2). We record the information in a square matrix A whose (i, j)-entry ai j is the number of ways that Ant i can reach Morsel j .