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Thread: No of ways ?

  1. #1
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    No of ways ?

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

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

    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?
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