A number has exactly 75 divisors if its prime factorization is one of these forms: p^74, p^2*q^24, p^4*q^14, p^2*q^4*r^4. The ways correspond to 75, 3*25, 5*15, 3*5*5 which are all 75. For every prime up to N, count the number of times it divides N!. Then you can do simple nested loops to check all possible choices for each pattern.