How many $$6 \times 7$$ matrices are there with entries in {0,1} such that all the row totals and column totals are odd numbers?
If the row sums are all odd, then the total number of 1s is the sum of these 6 odd numbers, hence even. If the column sums are all odd, then the total number of 1s is the sum of these 7 odd numbers, hence odd. A contradiction; hence the number of such matrices is 0
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