How many positive integers $x$ with $200 \leq x \leq 600$ have exactly one digit that is a prime number?
The prime digits are 2, 3, 5, and 7, and the non-prime digits are $0,1,4,6,8$, and 9 .

If the first digit is 2 , then the other two digits cannot be prime.

Since 6 of the digits are non-prime, there are $6 \times 6=36$ three-digit integers with first digit equal to 2 that have exactly one digit that is prime.

Similarly, with the first digit equal to each of 3 and 5, there are 36 three-digit integers with exactly one digit that is prime.

If the first digit is 4 , then exactly one of the other two digits must be prime.

There are $4 \times 6=24$ ways to choose a prime digit and a non-prime digit.

There are 2 choices for where to put the prime digit (the tens position or the units position).

Therefore, there are $2 \times 24=48$ three digit integers starting with 4 that have exactly one digit that is prime.

The answer is therefore $36+36+36+48=156$.