proof and power sets

[marble fossil]

I'm only 15, so take this with a grain of salt for the rigor expected of a uni course.
However, I think it represents ||the number of possible subsets|| because if you have n elements, you consider:

subsets with 0 elements, or n choose 0
subsets with 1 element, or n choose 1
subsets with 2 elements, or n choose 2
...
subsets with n-1 elements, or n choose n-1
subsets with n elements, or n choose n
adding these together, we get the sum thingy in part (b)

I just realized that spoiler was essentially useless cause I didn't spoiler the rest but oh well lol