$\displaystyle \\\textup{In how many ways can one fill a 4X4 matrix with }\pm1\\$
$\displaystyle \textup{so that the product of the entries in each row and each column is equal to -1 ?}$
Put a + in each column in such that no row has more than one +. $\displaystyle \begin{array}{*{20}c} - & + & - & - \\ - & - & - & + \\ + & - & - & - \\ - & - & + & - \\ \end{array} $.
That is one way to do this. And that can be one of $\displaystyle 4!=24$ ways.
Now, change each + to – and each – to plus. How many ways do you have?
Suppose we use 6 +’s and 10 –‘s. $\displaystyle \begin{array}{*{20}c} + & - & - & - \\ + & + & + & - \\ + & - & - & - \\ - & - & - & + \\ \end{array} $
How many ways can we rearrange the columns and rows? Be careful of duplications.
Again change the +’s and –‘s.
Is there another possible scheme?