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- Each of the 5 non-A positions can be labeled U, C, or G (3 choices each). So, number of labelings is $3^5 = 243$.
- Therefore, total number of valid sequences is:
- \times 243 = 4860
- Question: A science fair judge evaluates 7 student projects, each judged on creativity (C), scientific rigor (R), and presentation (P), with each project receiving a score in each category: high (H) or low (L). The judge considers a project Exceptionally Strong if it receives high scores in at least 5 of the 3 categories. Assuming each score is independently high with probability $ \frac{1}{2} $, what is the probability that exactly 3 out of the 7 projects are Exceptionally Strong?
- Solution: First, compute the probability that a single project is Exceptionally Strong, i.e., gets H in at least 5 of the 3 categories. But there are only 3 categories, so getting high in at least 5 is impossible.
- Wait — this is a contradiction. A project cannot get high scores in more than 3 categories. The condition at least 5 high scores is impossible.
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