The figure in Panel I below is a grid of cells with four rows and four columns. The numbers on the top and on the left represent the number of cells that are to be shaded in that column and row, respectively. Which one of the options shown in Panel II below represents the grid shaded correctly?
Show Hint
Start with the row that allows only one shaded cell; it eliminates most of the wrong options immediately.
Step 1: Note down the required totals.
Panel I is a 4 by 4 grid. The column totals, written across the top, are 2, 2, 2, 2, so every column must end up with exactly 2 shaded cells. The row totals, written down the side, are 3, 1, 2, 2 from top to bottom, so row 1 needs 3 shaded cells, row 2 needs 1, row 3 needs 2, and row 4 needs 2.
Step 2: Count the shaded cells in each row of option (i).
Reading option (i) row by row from top to bottom, the shaded counts are 3, 2, 1, 2. Row 1 matches the required 3, but row 2 has 2 shaded cells instead of the required 1, and row 3 has 1 shaded cell instead of the required 2. Since two rows already break the rule, option (i) is wrong.
Step 3: Count the shaded cells in each row of option (ii).
Reading option (ii) row by row, the shaded counts are 3, 1, 2, 2, which matches the required row totals of 3, 1, 2, 2 exactly. Now check the columns of option (ii): column by column the shaded counts come out to 2, 2, 2, 2, which also matches the required column totals. Every row and every column checks out.
Step 4: Rule out options (iii) and (iv).
Option (iii) has row counts of 3, 2, 1, 2, the same mismatch as option (i), so it fails on the row totals. Option (iv) has row counts of 3, 1, 1, 3, which does not match the required 3, 1, 2, 2 either, so it also fails.
Final Answer:
Only option (ii) shades exactly 3, 1, 2, 2 cells in the four rows and exactly 2, 2, 2, 2 cells in the four columns, matching Panel I.
\[ \boxed{\text{(ii)}} \]
