1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 16875. The answer should be presented as an integer.
Answer: 1120.

Solution. Since $16875=3^{3} \cdot 5^{4}$, the sought numbers can consist of the following digits: (a) three threes, four fives, and one one, or (b) one three, one nine, four fives, and two ones. We will calculate the number of variants in each case.

(a) First, we choose three places out of eight for the threes $\left(C_{8}^{3}=\frac{8!}{3!5!}\right.$ ways), then one place out of the remaining five for the one ( $C_{5}^{1}=5$ ways). Finally, the remaining places are occupied by fives. By the multiplication rule, we get $C_{8}^{3} \cdot C_{5}^{1}=\frac{8!}{3!5!} \cdot 5=280$ ways.

(b) Reasoning similarly, we find that the number of ways in this case is $\frac{8!}{4!2!}=840$. Finally, we get $280+840=1120$ ways.