Problem 13 · 2024 Math Kangaroo
Hard
Counting & Probability
caseworkcareful-counting
Jelena fills the 2×4 table shown with the letters A, B, C and D. She wants each letter to appear exactly once in each row and exactly once in each of the three 2×2 squares. In how many ways can she do this?
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Answer: B — 24
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Hint 1 of 2
Fill the left-hand 2x2 square first, then ask how much freedom is left.
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Hint 2 of 2
Because the 2x2 squares overlap by one column, knowing two columns almost forces the next.
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Approach: fill the first 2x2 block, then propagate the overlaps
- The left 2x2 block can be filled with the four letters in \(4!=24\) ways.
- Once columns 1 and 2 are set, the middle 2x2 block (columns 2-3) must contain the same four letters, so column 3 is forced to be column 1 with its two entries swapped between the rows; the same logic then forces column 4.
- So every choice of the first block extends to exactly one full table, giving \(24\) ways, answer B.
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