If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:
Find the number of ways to distribute 5 distinct employees into 4 indistinguishable offices, with empty offices allowed.
Concept Used:
When offices are indistinguishable, we are essentially partitioning the set of 5 distinct employees into at most 4 unlabeled subsets (since offices are indistinguishable). This is equivalent to counting the number of ways to partition a set of 5 distinct objects into at most 4 non-empty unlabeled subsets, which is given by the Bell number \(B_5\) minus the number of partitions into 5 subsets (which would require 5 offices, but we only have 4). However, since empty offices are allowed, we can use between 1 and 4 non-empty subsets.
Step-by-Step Solution:
Step 1: Interpret the problem.
We have 5 distinct employees and 4 indistinguishable offices. Any office can have any number of persons, including zero. This means we are partitioning the set of 5 employees into at most 4 unlabeled subsets (some offices may be empty).
Step 2: Count the number of partitions.
Let \(S(5,k)\) denote the Stirling numbers of the second kind (number of ways to partition 5 distinct objects into k non-empty unlabeled subsets).
We need number of partitions into 1, 2, 3, or 4 non-empty subsets (since we have 4 offices, we can use 1, 2, 3, or 4 of them).
Step 3: Recall Stirling numbers of the second kind for n=5:
Total number of partitions of 5 elements = Bell number \(B_5 = S(5,1)+S(5,2)+S(5,3)+S(5,4)+S(5,5) = 1+15+25+10+1 = 52\).
We exclude the case where all 5 are in separate offices (5 non-empty subsets) because that would require 5 offices, but we only have 4. So \(n = B_5 - S(5,5) = 52 - 1 = 51\).
