What are the elements of a partition S? Why, they are subsets of A. So if T is to be a subset of S, elements of T are subsets of A. Your T does not consist of subsets of A; e.g. 1 is an element of T, but 1 is certainly not an element of S.
You've almost solved your problem. Take T to be a 4 element subset of S with the union of the subsets of A in T totaling 10 integers.