Problem 16 · 2011 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
The cells of the 4×4 table shown should each be coloured either black or white. The numbers tell how many cells in each row and column should be black. In how many ways can the colouring be done?
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Answer: D — 5
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Hint 1 of 2
A row or column needing 0 black cells is entirely white, shrinking the puzzle.
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Hint 2 of 2
Count the black-cell placements left over for the smaller grid by cases on the row needing 2.
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Approach: remove the zero line, then enumerate the 3x3 core
- The row and column needing 0 black cells are all white, leaving a 3×3 core with row sums 2,1,1 and column sums 2,1,1.
- Place the two black cells of the '2' row, then fill the two '1' rows to meet the column totals.
- The three cases for that pair of columns give 2 + 2 + 1 = 5 valid fillings.
- So there are 5 ways, choice (D).
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