Problem:
How many different arrays whose entries are all 's and 's have the property that the sum of the entries in each row is and the sum of the entries in each column is
Solution:
Each row and each column must contain two 's and two 's, so there are ways to fill the first row. There are also six ways to fill the second row. Of these, one way has four matches with the first row, four ways have two matches with the first row, and one way has no matches with the first row. The first case allows one way to fill the third row, the second case allows two ways to fill the third row, and the third case allows six ways to fill the third row. Once the first three rows are filled, the fourth row can be filled in only one way. There are thus ways to fill the array to satisfy the conditions.
The problems on this page are the property of the MAA's American Mathematics Competitions