In how many ways can five people be divided into several groups?
(Number of member can be 1 to 5)
One person can't be in two groups at the same time.
Also, everyone have to join any groups.
I think, I should using the Stirling numbers of the second kind.
Is this right?
Let $(v_1,\ldots,v_n)$ define $n$ groups, with $v_i$ people in $i$'th group, where $5\geq n\geq1$ and $5\geq v_i>0$. We say $(v_1,\ldots,v_n)=m$ to define the $m$ ways this grouping can be performed. WLOG we assume that $v_i\leq v_{i+1}$.
For example: if we split 5 people into 3 groups then we have only these two options: $(1,1,3)$ and $(1,2,2)$.
These two are obvious $(1,1,1,1,1)=(5)=1$.
Notice that $(v_1, v_2)={5\choose v_1}={5\choose v_2}$, basically when we have only 2 groups, we need to calculate number of ways we can choose $v_1$ (or $v_2$) people out of 5.
Thus $(1,4)={5\choose 1}={5\choose 4}=5$, and $(2,3)={5\choose 2}={5\choose 3}=10$.
For $(1,1,3)$ we only look into all ways 3 people can be chosen out of 5, and same for $(1,1,1,2)$,
thus $(1,1,3)=(1,1,1,2)={5\choose 3}={5\choose 2}=10$.
Lastly, $(1,2,2)=5\times3=15$.
$\mathrm{TOTAL}=52$ ways.
Sorry for the mess up writing, I hope this might help you to get a general formula.