How many non-empty subsets of $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ consist entirely of odd numbers?
I notice that the set $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ has four odd numbers: 1, 3, 5, and 7. So, any non-empty subset of odd numbers must be a subset of $\{ 1, 3, 5, 7 \}$. How many subsets does $\{ 1, 3, 5, 7 \}$ have? I recall that the number of subsets of a set with n elements is $2^n$, since each element can be either included or excluded. So, $\{ 1, 3, 5, 7 \}$ has $2^4 = 16$ subsets. But wait, one of those subsets is the empty set, which I don't want to count. So, I need to subtract one from the total. Therefore, the number of non-empty subsets of odd numbers is $16 - 1 = 15$. 