Positive integers less than 50 are classified into two groups: perfect cubes less than 50, or integers formed by the product of exactly two distinct prime numbers. These numbers possess precisely two distinct factors besides 1 and themselves.
\(N = p^3\) (where \(p\) is a prime number)
\(N = p_1 \times p_2\) (where \(p_1\) and \(p_2\) are prime numbers)
In Case I, the perfect cubes of primes less than 50 are 8 and 27, totaling 2 numbers.
In Case II, numbers in the format \((2 \times 3), (2 \times 5), (2 \times 7), (2 \times 11), (2 \times 13), (2 \times 17), (2 \times 19), (2 \times 23), (3 \times 5), (3 \times 7), (3 \times 11), (3 \times 13), (5 \times 7)\) are identified, totaling 13 numbers.
Therefore, the total count of numbers with exactly two distinct prime factors is \((13 + 2) = 15\).