Let's define persistence of an integer number p(N) as the number of times one has to multiply said number's digits before reaching a single digit.
Examples: 48==>32==>6, so p(48)=2;
23568643407==>0 , so p(23568643407)=1

Three tasks:
a. (easy). Comment on the value of p(N!) for a non-negative N.
b.(harder) What can be said of the values of p(N) in bases 2 or 3.
c. (hardest) How many 3-digit integers with p(N)=3 are there?

Try to solve analytically.