Panel I below shows a grid of cells with four rows and four columns. The numbers on the top of each column and on the left of each row tell us how many cells in that column or row must be shaded. The column totals, left to right, are 2, 2, 2, 2, and the row totals, top to bottom, are 3, 1, 2, 2. Panel II below shows four candidate grids, labelled (i) to (iv), with some cells shaded grey.
Which one of the options shown in Panel II represents the grid shaded correctly?
Show Hint
Add up the shaded cells in each row and each column separately, and match both sets of totals against the numbers given in Panel I.
Step 1: Note down the required totals.
Reading Panel I, each of the \(4\) columns must have exactly \(2\) shaded cells, and the \(4\) rows, from top to bottom, must have \(3\), \(1\), \(2\), and \(2\) shaded cells respectively.
Step 2: Count the shaded cells in each row of every option.
For option (i), the rows have \(3\), \(2\), \(1\), \(2\) shaded cells. The required row totals are \(3\), \(1\), \(2\), \(2\), so rows \(2\) and \(3\) do not match. Option (i) is rejected.
For option (ii), the rows have \(3\), \(1\), \(2\), \(2\) shaded cells, which matches the required row totals exactly.
For option (iii), the rows have \(3\), \(2\), \(1\), \(2\) shaded cells, the same mismatch as option (i). Option (iii) is rejected.
For option (iv), the rows have \(3\), \(1\), \(1\), \(3\) shaded cells, which does not match the required \(3\), \(1\), \(2\), \(2\). Option (iv) is rejected.
Step 3: Confirm option (ii) also satisfies the column totals.
Counting down each column of option (ii) gives \(2\) shaded cells in every one of the \(4\) columns, which matches the required column total of \(2\) for each column.
Step 4: Final conclusion.
Only option (ii) satisfies both the row totals and the column totals at the same time.
\[
\boxed{\text{(ii)}}
\]
