In this section, there were four cases, the 4 combinations of distinguishable and identical balls and cells. I could understand [Distinguishable balls and cells] and [identical balls and distinguishable cells] but I just can not understand this one.
First of all, i don't understand stirling numbers of second kind, i had read a little about one of the types of stirling number while studying circular permutations but I was not very well versed with it (read: i am still confused). Secondly, i don't understand how we 'easily' get the two results they have shown.
I would really appreciate if someone could explain this to me and also a little bit of Sterling numbers.
Thanks!
The set $\{1,2,3\}$
can be partitioned into $\mathbf{three}$ subsets in $\mathbf{one}$ way: $\{\{1\},\{2\},\{3\}\}$ so $S(3,\mathbf{3})=\mathbf{1}$;
into $\mathbf{two}$ subsets in $\mathbf{three}$ ways: $\{\{1,2\},\{3\}\}$, $\{\{1,3\},\{2\}\}$, and $\{\{1\},\{2,3\}\}$, so $S(3,\mathbf{2})=\mathbf{3}$;
and into $\mathbf{one}$ subset in $\mathbf{one}$ way: $\{\{1,2,3\}\}$, $S(3,\mathbf{1})=\mathbf{1}$.
There are $\mathbf{zero}$ ways to partitioned $\{1,2,3\}$ into $\mathbf{four}$ non-empty subsets $S(3,\mathbf{4})=\mathbf{0}.$