Among 200 journalists, there are:
- 175 speak German
- 150 speak French
- 180 speak English
- 160 speak Japanese
Each journalist can speak at least one of the 4 languages. What is the maximum possible number of journalists who can speak all of them? What is the minimum possible number of journalists who can speak all of them?
Attempt:
For the maximum part, it should be $150$, because if there is more than 150, it means there are more than 150 who speak French which is not the fact.
From PIE
$$|G \cap F \cap E \cap J| = 465 - \left( |G \cap F| + |G \cap E| + | E \cap F| + |E \cap J| + |G \cap J| + |F \cap J| \right) + \left( |G \cap F \cap E| + |G \cap J \cap F| + |G \cap J \cap E| + |E \cap F \cap J| \right) $$
so to find the minimum i have to find the maximum of the $$ \left( |G \cap F| + |G \cap E| + | E \cap F| + |E \cap J| + |G \cap J| + |F \cap J| \right) + \left( |G \cap F \cap E| + |G \cap J \cap F| + |G \cap J \cap E| + |E \cap F \cap J| \right) $$
What is the idea?
For the minimum, you can observe that
Thus $25 + 50 + 20 + 40 = 135$ is the maximum number of journalists who don't speak at least one language (in the case where the 4 previous sets are disjoint). Thus the minimum number who speak the four languages is $200 - 135 = 65$.
EDIT. In more details, let us partition the set $S$ of 200 journalists in 5 disjoint subsets, $S_1$ to $S_5$, with $|S_1| = 25$, $|S_2| = 50$, $|S_3| = 20$, $|S_4| = 40$ and $|S_5| = 65$. Let now $E = S_1 \cup S_2 \cup S_4 \cup S_5$, $F = S_1 \cup S_3 \cup S_4 \cup S_5$, $G = S_2 \cup S_3 \cup S_4 \cup S_5$ and $J = S_1 \cup S_2 \cup S_3 \cup S_5$. Then $|E| = 180$, $|F| = 150$, $|G| = 175$, $|J| = 160$ and $|E \cap F \cap G \cap J| = |S_5| = 65$.