Let S={ a,b,c,d,e,f,g,h }
(a) There are 8 different subsets each containing 1 element.
(b) There are 8C2=8.7/(1.2)=28 different subsets each containing 2 elements:
{ a,b }, { a,c }, { a,d }, { a,e }, { a,f }, { a,g }, { a,h },
{ b,c }, { b,d }, { b,e }, { b,f }, { b,g }, { b,h }, { c,d },
{ c,e }, { c,f }, { c,g }, { c,h }, { d,e }, { d,f }, { d,g },
{ d,h }, { e,f }, { e,g }, { e,h }, { f,g }, { f,h }, { g,h }.
(c) 8C3=8.7.6/(1.2.3)=56 subsets of size 3.
(d) 8C4=8.7.6.5/(1.2.3.4)=70 subsets of size 4.
(e) There are 8+28+56+70+56+28+8=254=28-2 subsets of all sizes excluding the empty set and S itself.
The above are properties of Pascal's Triangle. The 8th row coefficients are:
1 8 28 56 70 56 28 8 1, sum to 28, which is all subsets including the empty set and S itself.