Problem 16 · 2017 Math Kangaroo
Hard
Counting & Probability
Logic & Word Problems
careful-counting
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 three days a week. How many different weekly plans meet those conditions?
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Answer: B — 7
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Hint 1 of 2
A weekly plan repeats, so the seven days form a loop — Sunday touches Monday.
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Hint 2 of 2
Count choices of 3 days on a circle of 7 with no two adjacent.
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Approach: count non-adjacent triples around a cycle of 7 days
- Because the plan repeats every week, the days form a circle of 7 where no two chosen days may be next to each other.
- The number of ways to pick 3 non-adjacent positions on a circle of 7 is 7.
- So there are 7 possible weekly plans, choice B.
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