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But wait: in circular arrangements, fixing one non-birch (to break rotational symmetry) and arranging the rest relative to it is equivalent to the above method. Since the grains are otherwise indistinct by type, and we are counting distinct circular configurations up to rotation, the above count is valid.
Question: A science educator designs a digital learning module where students complete 7 sequential interactive tasks, each tagged with one of three difficulty levels: Easy (E), Medium (M), and Hard (H). If each task is assigned a difficulty uniformly at random, what is the probability that exactly two tasks are Hard and no two Hard tasks are consecutive?
Solution: We are to compute the probability that, in a random assignment of difficulty levels (E, M, H) to 7 tasks, exactly 2 are Hard and no two Hard tasks are adjacent.

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