A simple proof on the number of (3 × n)-Latin rectangles based on a set of λ elements

In 1980, Athreya, Pranesachar and Singhi established the chromatic polynomial of (3 × n)-Latin rectangles whose entries based on a set \1, 2, ..., λ\ in which λ ≥ n. Their proof requires M\"{o}bius inversion formula and lattice partitions. In this paper, we present a simpler proof by using the idea of mathematical induction and appropriate coloring.