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Step 1: Understand the Problem
We have five distinct employees that need to be distributed into four offices. The offices are indistinguishable (identical), and there is no restriction on how many employees can be in each office—an office can even be empty.
Step 2: Identify the Concept (Stirling Numbers of the Second Kind)
When distributing distinct items (employees) into indistinguishable groups (offices), one can use the concept of Stirling numbers of the second kind. $S(n,k)$ counts the number of ways to partition $n$ distinct objects into $k$ non-empty identical groups.
Step 3: Formulate the Required Sum
Because we have up to 4 groups (offices) and an office can be empty, we need to consider all partitions of 5 employees into 1, 2, 3, or 4 non-empty groups. Symbolically, this is given by:
$$
S(5,1) + S(5,2) + S(5,3) + S(5,4).
$$
Step 4: Compute Each Stirling Number
$S(5,1) = 1$ (all 5 employees in one group).
$S(5,2) = 15$.
$S(5,3) = 25$.
$S(5,4) = 10$ (three groups get one employee each, and the remaining two employees go in the fourth group).
Step 5: Sum the Values
Add the values obtained in step 4:
$$
1 + 15 + 25 + 10 = 51.
$$
Step 6: Conclude the Final Answer
The number of ways to seat five distinct employees into four indistinguishable offices (with no restriction on occupancy) is
$$
\boxed{51}.
$$
