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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:
\( S(6, 1) = 1 \) (all artifacts in one bin)
\( S(6, 2) = 31 \) (two non-empty bins)
\( S(6, 3) = 90 \) (three non-empty bins)
\( S(6, 4) = 65 \) (four non-empty bins)
Summing these:
+ 31 + 90 + 65 = 187