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5Question: A lunar geological mapping robot collects 5 rock samples, 3 soil samples, and 2 mineral specimens. If it stores them in a sequence, how many distinct arrangements are possible if samples of the same type are indistinguishable?
Solution: The total number of samples is $5 + 3 + 2 = 10$. Since samples of the same type are indistinguishable, the number of distinct arrangements is given by the multinomial coefficient:
Question: A philosopher of science studies 4 biological theories and 3 philosophical frameworks. If they select one item per day for 7 days without replacement, how many distinct sequences of study are possible?
Solution: The philosopher has $4 + 3 = 7$ distinct items (treating each theory and framework as unique). The number of ways to arrange them is the number of permutations of 7 distinct objects:
Question: Three integers are selected uniformly at random from 0 to 50 inclusive. What is the probability that their sum is divisible by 3?
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