# Can this crate have even numbers in all rows and columns?

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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$$\begin{array}{ccc} B&B&B&.&.&B\\ B&B&B&.&B&.\\ B&B&B&B&B&B\\ B&B&B&B&.&. \end{array}$$
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