## Task 15/69

In how many different ways can the number 30030 be written as a product of three natural numbers (all different from 1) factors (where the order of the factors does not matter)?
The prime factorization of the given number 30030 is

$$
30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13
$$

The number 30030 thus contains 6 distinct prime factors, several of which need to be combined into a single factor. There are several possibilities for this:

1. Four prime factors are combined into one factor, and the two remaining prime factors form the second and third factors. From the 6 prime factors, 4 can be chosen in $\binom{6}{4}=\binom{6}{2}=15$ different ways.
2. Three prime factors are combined into the first factor, 2 more into the second factor, and the remaining prime factor forms the third factor. From the 6 prime factors, 3 can be chosen in $\binom{6}{3}=20$ different ways, and from the remaining 3 prime factors, 2 can be chosen in $\binom{3}{2}=3$ different ways, so the total number of factors that can be formed in this way is $20 \cdot 3=60$.
3. Two prime factors are combined into each factor. Analogous to the considerations in the second case, this can be done in $\binom{6}{4} \cdot\binom{4}{2}=90$ different ways. Since the three factors can be permuted in $3!=6$ different ways without yielding a new factorization, this number must be divided by 3!. The total number of products that can be formed in this way is thus $90: 6=15$.

Therefore, the total number of all possible representations of the number 30030 as a product of 3 natural factors (without considering the order) is $15+60+15=90$.