Assume that the set s has 8 elements. how many subsets of s have at most 4 elements?

Subsets having AT MOST 4 elements, means the subsets could have:
4 elements (at most)
3 elements
2 elements
1 element
0 element (nil element = Ф)

You can solve through combination; However remember that
ⁿCₓ = (n!)/(x!)(n-x)!

4 elements  = ⁸C₄ = 70
3 elements  = ⁸C₃ = 56
2 elements  = ⁸C₂ = 28
1 element    = ⁸C₁ =  8
Ф element   = ⁸C₀ = 1

Adding up gives 163 subsets with at most 4 elements

With a continuous thought process.