## Construction workers

### Task number: 4158

Pavel is hiring workers to build a new house. Each of the 80 workers who have applied for recruitment masters at least one of the professions: bricklayer, carpenter, painter, and even 15 of them control all three professions. Furthermore, Pavel found out that 50 people interested in work know how to lay walls and that the same number of painters are among those who are interested. Only 45 people control the carpentry trade.

How many workers would Paul hire if he chose all those who manage exactly two professions?

• #### Solution

Let us denote $$A, B, C$$ the sets of masons, painters and carpenters.

We know from the assignment that $$|A\cup B \cup C|=80, |A|=|C|=50, |B|=45, |A\cap B\cap C|=15$$.

By modifying the principle of inclusion and exclusion: $$|A\cup B \cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B \cap C|$$ we will get: $$|A\cap B|+|A\cap C|+|B\cap C|=|A|+|B|+|C|+|A\cap B \cap C|-|A\cup B \cup C|=50+45+50+15-80=80$$

In each of these three sets, however, there are 15 omniscients who control all three professions, so we must subtract them three times.

It would also be correct to verify that a set system with the stated properties exists at all. There are several such, for example: