Solution: This is equivalent to partitioning 4 distinguishable objects into 2 non-empty indistinct subsets, calculated by the Stirling number $ S(4, 2) $. Using the recurrence:

["Understanding the Solution: Partitioning 4 Distinguishable Objects into 2 Non-Empty Indistinct Subsets via Stirling Numbers", "When solving combinatorics problems involving partitioning objects, particularly those that are distinguishable or indistinguishable, Stirling numbers of the second kind provide a powerful mathematical tool. One classic example is computing the number of ways to partition 4 distinguishable objects into 2 non-empty, indistinct subsets, a problem succinctly expressed using the Stirling number of the second kind, denoted $ S(4, 2) $.", "This article explores how Stirling numbers calculate such partitions and walks through the recurrence relation that defines them, offering both theoretical clarity and practical calculation.", "---", "### What Does It Mean to Partition Distinguishable Objects into Indistinct Subsets?", "Consider 4 distinct items—say, apples labeled A, B, C, and D. A partition of these objects means dividing them into one or more groups, where each object belongs to exactly one group. When the subsets are indistinguishable, the order of subsets does not matter—i.e., grouping {A,B} and {C,D} is the same as {C,D} and {A,B}. A non-empty partition means no subset is empty.", "Thus, partitioning 4 distinguishable objects into 2 non-empty, indistinct subsets asks: In how many unique ways can I split these 4 labeled items into exactly two non-empty groups, without caring which group is first or second?", "---", "### The Stirling Number of the Second Kind: $ S(n, k) $", "The number of ways to partition $ n $ distinguishable objects into $ k $ non-empty, indistinct subsets is given by the Stirling number of the second kind, $ S(n, k) $. This function satisfies a recurrence relation that enables efficient computation:", "[\nS(n, k) = k \cdot S(n-1, k) + S(n-1, k-1)\n]", "This recurrence expresses how adding a new object creates new partitions:\n- Either the new object joins one of the existing $ k $ subsets (multiplied by $ k $, since the subsets are indistinct and order doesn’t matter),\n- Or it forms a new subset by itself, but only if $ k \geq 2 $ and we have at least one subset to join — hence the $ S(n-1, k-1) $ term when it starts a new group.", "Base cases:\n- $ S(n, 1) = 1 $ for all $ n \geq 1 $: all objects in one subset.\n- $ S(n, n) = 1 $: each object in its own subset.\n- $ S(n, k) = 0 $ if $ k > n $ or $ k = 0 $ (except $ S(0,0) = 1 $ conventionally).", "---", "### Calculating $ S(4, 2) $ Using the Recurrence", "To find $ S(4, 2) $, apply the recurrence step-by-step:", "1. Start with small values:\n - $ S(1,1) = 1 $\n - $ S(2,1) = 1 $, $ S(2,2) = 1 $\n - $ S(3,1) = 1 $,\n $ S(3,2) = 2 \cdot S(2,2) + S(2,1) = 2 \cdot 1 + 1 = 3 $\n (which matches known partitions: {A|B|C}, {A|C|B}, {B|C|A} — indistinct so order doesn’t matter).", "2. Compute $ S(4,2) $:\n $$\n S(4, 2) = 2 \cdot S(3, 2) + S(3, 1) = 2 \cdot 3 + 1 = 7\n $$", "---", "### Verifying by Direct Enumeration", "For confirmation, list all valid partitions of {A,B,C,D} into 2 non-empty, indistinct subsets:", "- {A,B} | {C,D}\n- {A,C} | {B,D}\n- {A,D} | {B,C}\n- {B,C} | {A,D}\n- {B,D} | {A,C}\n- {C,D} | {A,B}\n- {A,B,C} | {D}\n- {A,B,D} | {C}", "Under indistinct subsets, the first 7 listed (e.g., {A,B}|{C,D} and {C,D}|{A,B} are identical) yield exactly 7 unique groupings, matching $ S(4,2) = 7 $.", "---", "### Summary", "- Partitioning 4 distinguishable objects into 2 non-empty indistinct subsets is computed via $ S(4,2) $.\n- Using the recurrence $ S(n, k) = k \cdot S(n-1, k) + S(n-1, k-1) $:\n $$\n S(4, 2) = 2 \cdot S(3, 2) + S(3, 1) = 2 \cdot 3 + 1 = 7\n $$\n- This reflects the number of unique groupings without regard to subset order.", "Stirling numbers elegantly bridge combinatorial reasoning and computable formulae, making them indispensable in discrete mathematics, probability, and algorithm design.", "---", "Related keywords:\nStirling number of the second kind, partitioning objects, non-empty subsets, indistinct groups, combinatorics, recurrence relation, distributing items, labeled partitions", "Meta description:\nDiscover how Stirling numbers of the second kind compute the number of ways to partition 4 distinguishable objects into 2 non-empty indistinct subsets using the recurrence $ S(n,k) = k \cdot S(n-1,k) + S(n-1,k-1) $. Compute $ S(4,2) = 7 $ with clear examples and method."]