A) The number of different subsets of 4 elements is the number of 4-combinations from a given set {a,b,c,d,e,f,g} of 7 elements:

C(7, 4) = 7!/(4!(7 - 4)!)

C(7, 4) = 5*6*7/(2*3)

C(7, 4) = 35

There are 35 different subsets containing exactly 4 elements.

B) The total number of different subsets is:

C(7, 0) + C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7)

Since

C(7,0) = C(7, 7) = 1

C(7, 1) = C(7, 6) = 7

C(7, 2) = C(7, 5) = 21

C(7, 3) = C(7, 4) = 35

we have

C(7, 0) + C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7) = 2*(1 + 7 + 21 + 35) = 128

Including the empty set, the total number of different subsets is 128.