[Permutation and combination] Plug-in method If a unit subscribes to 30 pieces of information and sends them to 3 departments, each...

The core challenge in this distribution problem is to determine the number of ways to allocate 30 distinct pieces of information to three distinct departments, given the constraint that each department must receive at least one piece. This is a classic application of the inclusion-exclusion principle, but the "plug-in method" referenced likely denotes a combinatorial model where the items to be distributed are treated as distinct entities placed into distinct groups, with the groups having no upper capacity limit. The direct mathematical model is that of surjections: counting the number of onto functions from a set of 30 distinct elements (information pieces) to a set of 3 distinct elements (departments). The total number of unrestricted functions is \(3^{30}\), as each of the 30 pieces can be sent to any of the 3 departments. From this, we must subtract the functions that leave at least one department empty.

Applying inclusion-exclusion provides the precise count: \(3^{30} - \binom{3}{1} \times 2^{30} + \binom{3}{2} \times 1^{30}\). The term \(\binom{3}{1} \times 2^{30}\) accounts for all functions where a specific department is chosen to be empty, and the pieces are sent only to the remaining two. However, this subtracts the cases where two departments are empty (i.e., all pieces go to one department) twice, once for each choice of the single empty department. Therefore, we add back \(\binom{3}{2} \times 1^{30}\) to correct for this over-subtraction. The final combinatorial expression is thus \(3^{30} - 3 \times 2^{30} + 3\). This result is specific and absolute; it does not depend on the order of distribution or any prioritization among the departments, as long as both the items and the departments are distinguishable.

The practical implication of this calculation extends beyond a textbook exercise. In an organizational context, such as a corporate intelligence unit or a news desk, ensuring that all departments receive at least some information is a fundamental requirement for operational equity and functional coverage. The sheer magnitude of the result—\(3^{30}\) is approximately \(2.05 \times 10^{14}\), and the final count remains astronomically large—highlights the explosive number of possible allocation strategies. This combinatorial explosion underscores the critical need for automated, rule-based distribution systems in real-world information management, as manual allocation would be infeasible. The calculation also implicitly defines the boundary of the problem: it assumes each piece of information is treated as a unique, indivisible unit and that the departments are independent recipients with no further substructure or capacity constraints.

A common point of confusion in such problems is misapplying formulas for combinations or distributions of identical objects. The plug-in method's power here lies in its direct modeling of distinctness. An alternative, equivalent approach is to use Stirling numbers of the second kind, \(S(30,3)\), which counts the ways to partition the 30 items into 3 non-empty unlabeled subsets, and then multiply by \(3!\) to account for assigning these subsets to the three distinct departments: \(3! \times S(30,3)\). This yields the same numeric result, confirming the mechanistic logic. The primary analytical takeaway is that the constraint "each department receives at least one" transforms a simple exponential calculation into a more structured combinatorial count, which is fundamental for modeling fair or complete resource allocation in systems with multiple distinct actors and resources.