How many five-digit numbers can be written with these digits:

(1) $0,1,2$;

(2) $1,1,0,2,3$.
( $1^{\circ}$ ) The number of five-digit numbers that can be written with the digits $0,1,2$ is obtained by subtracting the number of groups starting with $0$ from the number of all 5-permutations of 3 elements. The number of such groups is equal to the number of 4-permutations of 3 elements, and thus the sought number is:

$$
v_{5}^{i}(3)-v_{4}^{i}(3)=3^{5}-3^{4}=162
$$

$\left(2^{\circ}\right)$ From the number of permutations that can be formed from the given 5 elements, we subtract the number of groups that start with $0$:

$$
\frac{5!}{2!}-\frac{4!}{2!}=60-12=48
$$

(László Bánó, Budapest.)

The problem was also solved by: A. Haar, P. Messer, L. Pető, J. Sonnenfeld, the V. district high school math circle.