Exercise · ZW-BELL-NUMBER-009 · ★★★ Advanced
Show that \(B_5=52\) by summing \(S(5,k)\) for \(k=1,\dots,5\), knowing that \(S(5,1)=1\), \(S(5,2)=15\), \(S(5,3)=25\), \(S(5,4)=10\) and \(S(5,5)=1\). Which step gives the correct result?
Choose one option.
A Add \(S(5,k)\) only up to \(k=3\), obtaining 41 B Add only the even \(k\), obtaining 26 C Add \(1+15+25+10+1=52\) D Multiply the values, obtaining 3750 Show answer C. Add \(1+15+25+10+1=52\)← Back to the question
Explanation The full sum of the five Stirling numbers gives \(1+15+25+10+1=52\), which is \(B_5\). Multiplying the values makes no sense because \(B_n\) is a sum over \(k\).
The Bell number \(B_n\) is the number of partitions of a set with \(n\) elements, that is, of ways to split it into nonempty disjoint subsets .
\[B_n=\sum_{k=0}^{n-1}\binom{n-1}{k}B_k\]
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