Probability. Not for the faint of heart

Let S be a given set. If, for some k>0, S1,S2,....Sk are mutually exclusive nonempty subsets of S such that union (i=1 to k) Si=S,

, then we call the set {S1,S2,…,Sk} a partition of S. Let Tn denote the number of different partitions of {1,2,…,n}. Thus,T1=1(the only partition being S1={1}), and T2=2(the two partitions being {{1,2}},{{1},{2}}).

(a)

Show, by computing all partitions, that T3=5,T4=15.

(b) Show that

T(n+1)=1+summation(k=1 to n) (nchoose k)*Tk

I honesty have no clue what this question is talking about and the lecture on the topic was pretty much her just saying 0<P(E)<1 and other easy stuff, only to get this question for homework. Please help!

Quick question: how can I use this formula?

I have the formula (Tn+1)=1+summation(k=1 to n)(n choose k)*Tk

I need to compute T10, the number of different partitions of {1,2,....,10}

I don't really know what to plug in other than n=10 and I don't really know how to compute the result.

Thank you.