No Counting

To count all possible two-digit numbers for which the units digit is not a prime number and the tens digit is a prime number, let's find the number of possibilities for each digit. Recall the prime numbers between $0$ and $9$ inclusive are $2,3,5,7$ . Hence, there are $4$ numbers between $0$ and $9$ that are prime and $6$ that are not prime.

So, there are $6$ possible numbers for the ones digit and $4$ possible numbers for the tens digit. By the fundamental counting principle, we can simply multiply these values to find the total number of two-digit positive integers for which the unit digit is not a prime number and the tens digit is a prime number:

$6 \cdot 4=24$ , there are $\boxed{24}$ numbers that satisfy our criteria.