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?
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Count shaded cells in each row and column of every option and match them against the given totals.
Step 1: Understanding the Question: Panel I is a 4 by 4 grid, and the numbers on top of each column and to the left of each row tell us exactly how many cells in that column and in that row must be shaded once the grid is completed. The row totals from top to bottom are 3, 1, 2, 2, and every one of the four columns must have exactly 2 shaded cells. We need to find which of the four candidate grids in Panel II satisfies every one of these row and column totals simultaneously. Step 2: Key Formula or Approach: The most reliable way to check each candidate is to count the shaded cells in every row and compare that count against the required row total, and then separately count the shaded cells in every column and compare against the required column total of 2. A candidate is correct only if it passes both checks, for all four rows and all four columns, at the same time. Step 3: Detailed Explanation: Checking option (ii): Row 1 has cells shaded in columns 1, 3 and 4, which is 3 shaded cells, matching the required row total of 3. Row 2 has a single shaded cell in column 2, matching the required row total of 1. Row 3 has shaded cells in columns 3 and 4, giving 2 shaded cells, matching the required row total of 2. Row 4 has shaded cells in columns 1 and 2, giving 2 shaded cells, matching the required row total of 2. So all four row totals are satisfied. Now checking the columns of option (ii): column 1 is shaded in row 1 and row 4, giving 2 shaded cells, matching the requirement of 2. Column 2 is shaded in row 2 and row 4, giving 2 shaded cells, matching the requirement of 2. Column 3 is shaded in row 1 and row 3, giving 2 shaded cells, matching the requirement of 2. Column 4 is shaded in row 1 and row 3, giving 2 shaded cells, matching the requirement of 2. All four column totals are also satisfied, so option (ii) fits the grid perfectly. Checking the remaining options briefly, option (i) shades 2 cells in row 2 instead of the required 1, so it violates the row 2 total. Options (iii) and (iv) also each shade an incorrect number of cells in at least one row or column when checked against the required totals of 3, 1, 2, 2 for rows and 2, 2, 2, 2 for columns, so they fail the test as well. Step 4: Final Answer: Only option (ii) has every row and every column total matching the numbers given in Panel I. \[ oxed{ ext{Option (ii)}} \]
