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2Question: A survey collects opinions from 100 participants, each answering Yes, No, or Unsure to a key question. How many distinct response patterns are possible if exactly 40 participants say Yes, 35 say No, and 25 say Unsure?
Solution: We are asked to count the number of distinct ways to assign responses to 100 participants such that 40 say Yes, 35 say No, and 25 say Unsure. Since the participants are distinguishable but we are only counting the number of distinct groupings (i.e., how many ways to divide the 100 people into three labeled groups of fixed sizes), this is a multinomial coefficient problem. The number of such response patterns is given by:
This counts the number of ways to assign the labels while accounting for indistinct counts within each category.

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