How many partitions of the set $\{1,2,3, \ldots , 100\}$ are there such that both
a) there are exactly three parts and
b) elements $1,2,3$ are in different parts.
Any help on this question would be fine. Hints are welcomed
2026-03-28 13:28:06.1774704486
Partitions of a set into three parts
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Hint: It's the same as the three times the number of ways to partition the set $\{4,5, \dots, 100\}$ into three different parts where as many as two of the parts are allowed to be empty (can you see why?), and a formula for the latter value exists (specifically, you should look into Stirling numbers of the second kind and modify the count to account for the possibly-empty sets).