Problem 19 · 2017 Math Kangaroo
Medium
Counting & Probability
careful-countingcasework
Tycho plans his running training. Each week he wants to go for a run on the same weekdays. He never wants to go for a run on two consecutive days, but he wants to go for a run two days a week. How many different weekly plans meet those conditions?
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Answer: B — 14
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Hint 1 of 2
The seven weekdays form a cycle; pick 2 run-days with no two next to each other.
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Hint 2 of 2
Count all pairs of 7 days, then remove the adjacent ones.
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Approach: choose 2 non-adjacent days on a 7-cycle
- Choosing 2 of 7 days gives C(7,2) = 21 pairs.
- Of these, 7 pairs are consecutive days, which are not allowed.
- Valid plans = 21 − 7 = 14.
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