Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553.
1. Identify the single-digit prime numbers. The single-digit prime numbers are \(2, 3, 5,\) and \(7\).
2. Choose 3 out of these 4 prime numbers to form a three-digit number. The number of ways to choose 3 out of 4 is given by the binomial coefficient:
   \[
   \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4
   \]
3. For each selection of 3 prime numbers, determine the number of ways to arrange them. Since we are arranging 3 distinct digits, the number of permutations is:
   \[
   3! = 6
   \]
4. Multiply the number of ways to choose the digits by the number of ways to arrange them:
   \[
   \binom{4}{3} \times 3! = 4 \times 6 = 24
   \]

Thus, the number of three-digit positive integers where the digits are three different prime numbers is \(24\).

The final answer is \(\boxed{24}\)