Problem 16 · 2018 Math Kangaroo
Hard
Counting & Probability
casework
From the list 1, 2, 3, 4, 5, 6, 7, Monika chooses 3 different numbers whose sum is 8. From the same list Daniel chooses 3 different numbers whose sum is 7. How many of the numbers were chosen by both Monika and Daniel?
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Answer: C — 2
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Hint 1 of 2
List the sets of three different numbers from 1-7 that sum to 7, then those that sum to 8.
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Hint 2 of 2
Daniel's sum 7 has only one possible set; compare it with each of Monika's options.
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Approach: enumerate the possible triples and compare
- Three different numbers from 1-7 summing to 7 can only be {1, 2, 4}, so that is Daniel's set.
- Summing to 8 gives Monika either {1, 2, 5} or {1, 3, 4}.
- Comparing {1, 2, 4} with {1, 2, 5} shares {1, 2}; with {1, 3, 4} shares {1, 4}.
- Either way exactly 2 numbers are common to both.
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