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Home » No integer solution? But problem implies one. - Richter Guitar

No integer solution? But problem implies one. - Richter Guitar

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 To solve the problem of forming a council of 3 members from a group of 7 individuals where the roles are indistinguishable, we need to calculate the number of combinations of 7 individuals taken 3 at a time. This is given by the binomial coefficient:

📌 inom{7}{3} = rac{7!}{3!(7-3)!} = rac{7 imes 6 imes 5}{3 imes 2 imes 1} = 35

📌 Since the roles are indistinguishable, we simply count the distinct groups of 3. Therefore, the number of different councils that can be formed is:

📌 A researcher is analyzing the distribution of 6 unique artifacts into 4 indistinguishable storage bins. How many distinct ways can the artifacts be distributed if each bin can hold any number of artifacts?

📌 This is a classic problem of distributing distinguishable objects (artifacts) into indistinguishable bins. The number of ways to distribute \( n \) distinguishable objects into \( k \) indistinguishable bins is given by the sum of Stirling numbers of the second kind, \( S(n, k) \), summed for \( r = 1 \) to \( \min(n, k) \):

📌 \sum_{r=1}^{\min(6,4)} S(6, r)

📌 We calculate each relevant Stirling number:

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