2018 AMC 10B Problem 5

"At least one prime" covers many different cases. The opposite, "no primes at all", is just one case, so count that and subtract.

1. All subsets. Each of the $8$ numbers is either in the subset or not, so there are $2^8 = 256$ subsets.
2. Sort the numbers. The primes are $2, 3, 5, 7$. The numbers that are not prime are $4, 6, 8, 9$.
3. Subsets with no prime. These use only $4, 6, 8, 9$. Each of these four is in or out, so there are $2^4 = 16$ of them. (This includes the empty set, which has no prime in it.)
4. Subtract. Every other subset has at least one prime:
$$256 - 16 = 240.$$

The answer is $\boxed{\textbf{(D)}\ 240}$.

The words "at least one" are a signal to count the opposite. The subsets with none of the special items are easy to count, and taking them away from the total leaves exactly the ones you want.

The set has $8$ elements, so it has $2^8 = 256$ subsets in total.

The primes in the set are $2, 3, 5, 7$; the non-primes are $4, 6, 8, 9$. A subset contains no prime exactly when it is a subset of $\{4, 6, 8, 9\}$, and there are $2^4 = 16$ of those (including the empty set).

Every other subset contains at least one prime:
$$256 - 16 = 240.$$

The answer is $\boxed{\textbf{(D)}\ 240}$.