How many different primes appear in the prime factorization of $20!$ (20 factorial)? (Reminder: The number $n!$ is the product of the integers from 1 to $n$.  For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.)
When we multiply the numbers from 1 to 20, we include every prime less than 20, and no numbers with prime factors greater than 20.  So, the prime factorization of $20!$ includes all primes less than 20, and no other primes.

The primes that fit this are $\{2,3,5,7,11,13,17,19\}$, and there are $\boxed{8}$ of them.