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A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?

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1 Answer

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Yes.

B B B . . B

B B B . B .

B B B B B B

B B B B . .

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That is one of many answers. I finally figured out several options proving the claim to be true. Thanks for your input! –  Jared Oct 3 '12 at 21:02
    
A more interesting problem comes if you generalize it: (1) under your constraints that the row and column entries are all even, under what conditions on the size of the crate and the number of bottles is a solution possible? (2) if a solution is possible, how many different (after you've defined "different") solutions are possible? –  Rick Decker Oct 3 '12 at 21:15
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Note that this is equivalent to picking six (empty) spaces in the grid so that the number of empty spaces in each row and column is even. If you have four empty spaces in one row you have only two left and can't make all the columns even. So the six spaces have to be distributed as pairs in three columns and three rows. So choose three columns and three rows. This gives nine points of intersection, of which we need just six. There are six configurations which work. So the number of possible arrangements is $6\binom 4 3 \binom 6 3$ –  Mark Bennet Oct 3 '12 at 22:49
    
So @Mark, should I ask my generalization as a new question so you can have a chance to shine? –  Rick Decker Oct 4 '12 at 1:04
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