21. Patricia painted some of the cells of a $4 \times 4$ grid. Carl counted how many red cells there were in each row and in each column and created a table to show his answers. Which of the following tables could Carl have created?
Solution
D

The top row of the table in option A indicates that all four cells in that row need to be painted but this is not possible if the left hand column is to have no cells painted. In each table, the sum of the row totals and the sum of column totals represents the total number of cells painted. The table in option B has a row total sum of 7 and a column total sum of 8 , so this is not a possible table. The table in option $\mathrm{C}$ indicates no coloured cells in the bottom two rows which contradicts the three coloured cells in the second column. Hence table $\mathrm{C}$ is not possible. The table in option E indicates no coloured cells in the top row but three coloured cells in each of the second and fourth columns which contradicts the one coloured cell in the fourth row. Hence table $\mathrm{E}$ is not possible.
Therefore, the only table that Carl could have created is the table in option D. One possible arrangement that would give this table is shown in the diagram.