I have a set $A:=\{1,2,3,4,5,6,7\}$.
How many ways I have to partition $A$ in $5$ non-empty subsets? Which are these ways?
2026-03-27 05:54:59.1774590899
How many ways to partition a set?
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You have to consider two cases: $7=1+1+1+1+3$, $7=1+1+1+2+2$.
Hence the number of such partitions is $$\binom{7}{3}+\frac{1}{2}\left(\binom{7}{2}\cdot\binom{5}{2}\right)=140.$$
If you are interested in the number of partitions of a set of $n$ elements in $k$ subsets then take a look here: https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind