Question 30

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?

Solution

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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