More paint, more grid

This is a simple application of Latin squares, a concept which I have recently learned from Mr. Mark Hennings.

If we assign the colors a number from 1-5, then we can see that our problem looks for the number of arrangements of squares such that no number occurs on any row, or column twice.

This number is given by the formula

$N = n!(n-1)! L(n,n)$

where $n$ represents the number of rows/columns in the grid, and $L(n,n)$ is the number of "normalized" Latin squares of order $n$ . A normalized Latin square is one whose elements in the first row and column are ordered as $1,2...n$ .

Plugging in $n=5$ , we get

$N = 5!4! L(5,5)$

And since $L(5,5) = 56$ , we see that $N= 120 \times 24 \times 56 = \boxed{161280}$