The 4! comes from the 4 suits, its correct.

The first card can be chosen from 4 suits, the 2:nd from 3 suits, the 3:rd from 2, and the last from just one suit.

The multiply theroem then gives us 4x3x2x1=4!

The formula calculates the # of badugi

hands exactly n high.

If you want to calculate the total # of badugi hands that are n high or better, then the formula would be:

(n choose 4)4!