How many two-digit prime numbers can be formed by choosing two different digits from the set $\{2, 7, 8, 9\}$ to be used as the tens digit and units digit?
We immediately note that if 2 is the units digit, then the number is composite. If 2 is the tens digit, only 29 is prime. If 7 is the tens digit, only 79 is prime. If 8 is the tens digit, only 89 is prime. Finally, if 9 is the tens digit, only 97 is prime. Thus, $\boxed{4}$ two-digit primes can be formed.