Problem 28 · 2018 Math Kangaroo
Stretch
Counting & Probability
caseworkdivisibility
Each number of the set {1, 2, 3, 4, 5, 6} is written into exactly one cell of a 2 × 3 table. In how many ways can this be done so that the sum of the numbers in every column and every row is divisible by 3?
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Answer: D — 48
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Hint 1 of 2
Group the numbers by remainder mod 3: {3,6} give 0, {1,4} give 1, {2,5} give 2.
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Hint 2 of 2
Decide the remainder pattern of the table first, then count how to place the actual numbers.
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Approach: work modulo 3 on the remainder pattern, then fill in
- By remainder mod 3 the numbers are two 0's (3,6), two 1's (1,4) and two 2's (2,5).
- Each row of three and each column of two must sum to a multiple of 3; find the valid remainder patterns.
- For each valid pattern, the two numbers in each remainder class can be swapped, and counting all arrangements gives 48.
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