Problem 27 · 2021 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
In the 4×4 table, some cells must be painted black. The numbers next to and below the table show how many cells in that row or column must be black. In how many ways can this table be painted?
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Answer: D — 5
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Hint 1 of 2
The row and column labelled 0 are entirely white, shrinking the puzzle to a 3×3 grid.
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Hint 2 of 2
Count black-cell placements in that 3×3 grid with row sums 2,2,1 and column sums 2,2,1.
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Approach: reduce by the zero row/column, then enumerate
- Row 2 and column 2 each need 0 black cells, so they are all white.
- What remains is a 3×3 grid needing row sums 2,2,1 and column sums 2,2,1.
- Enumerating where the single-black row and the two double-black rows go yields exactly five valid fillings.
- So there are 5 ways, choice (D).
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