A simple proof on the number of $(3\times n)$-Latin rectangles based on a set of $\lambda$ elements

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