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?

Tue, 05 May 2026 22:21:23 +0000

**Why 100-Participant Opinion Studies Are Generating Curiosity — And What the Numbers Really Reveal** In an era where data shapes public conversation, narrow insights from focused surveys are emerging as quiet signals of shifting attitudes. The question at the heart of many recent discussions: How many unique response patterns can emerge from a group of 100 participants, each answering “Yes,” “No,” or “Unsure” to a single key question? With a study reporting 40 Yes, 35 No, and 25 Unsure, the conversation shifts from raw data to insight: what does diversity in opinions really mean in a large sample? This particular survey, centered on a core question about behavior, trust, or preference, reflects more than just personal views—each response pattern reveals a dynamic snapshot of public sentiment. With exactly 40 agreeing, 35 rejecting, and 25 withholding final judgment, the distribution points to a nuanced landscape of ambiguity, alignment, and cautious curiosity. Such precise breakdowns are not only statistically meaningful but increasingly relevant as consumers and researchers seek clarity in a fragmented information landscape. Understanding how many distinct response combinations exist deepens appreciation for what these patterns represent. Using combinatorial math, the number of distinct response patterns where 100 participants choose among three options—Y/N/U with exact counts of 40, 35, and 25—is calculated as a multinomial coefficient: 100! / (40! × 35! × 25!). Though large and not praedicable in daily usage, this formula underscores the precision of survey design and the richness of audience diversity behind a seemingly simple tally.