There is an $n\times(n-1)$ matrix, the elements of it are $0$ or $1$. Each row is different. There is only one column with one $0$ or one $1$. How many such matrices are there? I got this question from Game WOW, Maze patterns in the fog of the woods of Seth. Players must find one "special" image from four images.
For example, we choose picture 3 because it is empty while others are solid. 
For another example, we choose picture 4 because it is a leaf while others are flowers.
I code for a program and get there are $24$ combinations of pictures suit the rule. And I have put the sequences into OEIS but I can't find the answer. Is there a formula that can calculate how many images there are when there are $n$ images?
Sorry for my poor English. I will edit my words if I break the rule of MSE.