Problem 30 · 2020 Math Kangaroo
Stretch
Counting & Probability
careful-countinggrid-counting
On the 8 × 8 board shown, in how many ways can you place two chips, one green and one red, on differently coloured cells, so that the chips are not in the same row or in the same column of the board?
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Answer: E — 1536
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Hint 1 of 2
Green and red must sit on opposite colours; count the two colour-orders separately.
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Hint 2 of 2
For a fixed green cell, count the opposite-colour cells that avoid its row and column.
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Approach: count valid opposite-colour pairs by removing the shared row/column cells
- Put green on a white cell (32 ways); a black cell shares its row or column in 4+4 = 8 cases.
- So 32 − 8 = 24 black cells work, giving 32·24 = 768 ordered pairs.
- By symmetry green-black/red-white gives another 768.
- Total = 1536, so the answer is 1536.
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