4. We call a subset $M$ with three elements of the set $A=\{1,2,3, \ldots, 10\}$ of "type $Z$ " if the product of the elements of the set $M$ has the last digit 0. Determine the number of subsets of $A$ of "type $Z$ ".

Daniela and Nicolae Stănică, G.M.
## Solution.

Let $M=\{a, b, c\}$.

Case I: $a=5$ $3 p$

The product of elements $b$ and $c$ must be an even number.

1) If $b=2$, then $c \in\{1,3,4,6,7,8,9,10\}$, which gives 8 subsets.
2) If $b=4$, then $c \in\{1,3,6,7,8,9,10\}$, which gives 7 subsets.
3) If $b=6$, then $c \in\{1,3,7,8,9,10\}$, which gives 6 subsets.
4) If $b=8$, then $c \in\{1,3,7,9,10\}$, which gives 5 subsets.

In total, for this case, there are 26 subsets.

Case II: $a=10$ $3 p$

The product of elements $b$ and $c$ can be either even or odd. Out of the 36 possible subsets for $a=10$, we eliminate the subsets that have already been used in Case I. There are 4 common sets, $\{5,2,10\},\{5,4,10\},\{5,6,10\},\{5,8,10\}$.

In total, for this case, there are 32 subsets.

The number of subsets that satisfy the problem's conditions is $26+32=58$ subsets. $1 \mathrm{p}$