# Thread: No of ways ?

1. ## No of ways ?

$\\\textup{In how many ways can one fill a 4X4 matrix with }\pm1\\$
$\textup{so that the product of the entries in each row and each column is equal to -1 ?}$

2. Originally Posted by banku12
$\\\textup{In how many ways can one fill a 4X4 matrix with }\pm1\\$
$\textup{so that the product of the entries in each row and each column is equal to -1 ?}$

This doesn't seem a hard problem, does it? It can be long and annoying, but the algorithm to follow seems very clear: in every row-column there must be either 1 or three -1's...well, try some combinations by yourself .

Tonio

3. Originally Posted by banku12
$\\\textup{In how many ways can one fill a 4X4 matrix with }\pm1\\$
$\textup{so that the product of the entries in each row and each column is equal to -1 ?}$

Put a + in each column in such that no row has more than one +. $\begin{array}{*{20}c} - & + & - & - \\ - & - & - & + \\ + & - & - & - \\ - & - & + & - \\ \end{array}$.
That is one way to do this. And that can be one of $4!=24$ ways.
Now, change each + to – and each – to plus. How many ways do you have?

Suppose we use 6 +’s and 10 –‘s. $\begin{array}{*{20}c} + & - & - & - \\ + & + & + & - \\ + & - & - & - \\ - & - & - & + \\ \end{array}$
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?