Solution: We are partitioning $6$ distinguishable objects (proposals) into $3$ non-empty, indistinct subsets (teams). This is given by the Stirling number of the second kind $S(6,3)$.

["Partitioning Proposals: How Stirling Numbers of the Second Kind Help Divide Distinguishable Ideas into Non-Empty Teams", "When faced with a collection of unique proposals—say, six innovative project ideas—managing and organizing them into effective, collaborative units is a common challenge. How can organizations systematically assign these distinguishable proposals into distinct, cohesive teams while ensuring no team remains empty? This problem is elegantly solved using a fundamental concept from combinatorics: the Stirling numbers of the second kind, specifically denoted ( S(n, k) ).", "## What Are Stirling Numbers of the Second Kind?", "The Stirling number of the second kind, ( S(n, k) ), represents the number of ways to partition ( n ) distinguishable objects into exactly ( k ) non-empty, unlabeled (indistinct) subsets. In our context, these “objects” are proposals, and the “subsets” are the collaborative teams formed without any empty groups.", "For example, dividing 6 distinguishable proposals into 3 non-empty, indistinct teams corresponds directly to computing ( S(6, 3) ).", "## Why Use ( S(6, 3) ) in Team Formation?", "Suppose you have six unique proposals, labeled for clarity but treated as interchangeable once grouped: say Proposal A, B, C, D, E, and F. The goal is to divide them into 3 meaningful, non-empty teams—teams without hierarchy or distinct names—where team labels don’t matter because, organizationally, team identity is fluid or unimportant.", "Using ( S(6, 3) ) answers the question: how many distinct ways can we group six distinguishable items into three non-empty teams? Each way captures a unique structural arrangement of collaboration, ensuring no team is empty and the teams themselves are indistinct (e.g., Team 1: {A, B}, Team 2: {C, D}, Team 3: {E, F} is the same as Team 3: {E, F}, Team 1: {A, B}, Team 2: {C, D}).", "## Computing ( S(6, 3) )", "Stirling numbers of the second kind can be computed recursively using:", "[\nS(n, k) = k \cdot S(n-1, k) + S(n-1, k-1)\n]", "with base cases:\n- ( S(n, 0) = 0 ) for ( n > 0 ),\n- ( S(0, k) = 0 ) for ( k > 0 ),\n- ( S(0, 0) = 1 ),\n- ( S(n, n) = 1 ) (each object in its own subset).", "Using this recurrence or lookup tables:", "[\nS(6, 3) = 90\n]", "Thus, there are 90 distinct ways to partition six distinguishable proposals into three non-empty, indistinct teams.", "## Practical Implications", "This combinatorial insight is valuable for organizations managing team-building, project allocation, or collaborative research. Rather than manually searching through 90 configurations, planners can use ( S(6, 3) ) to:", "- Systematically model team distribution\n- Guarantee balanced and non-empty team composition\n- Leverage mathematical rigor in operational design\n- Compare or optimize different grouping strategies", "## Summary", "Partitioning six distinguishable proposals into three non-empty, indistinct teams is a classic application of the Stirling number of the second kind. The value ( S(6, 3) = 90 ) tells us there are 90 unique, unlabeled ways to form such teams—ensuring structural fairness and completeness. By understanding and applying Stirling numbers, organizations gain a powerful tool for optimizing collaboration and resource division.", "---", "Keywords: Stirling numbers of the second kind, S(6,3), partitioning problems, combinatorics, non-empty subsets, indistinct teams, team formation, distinguishable objects, organizational design, project team division."]