How many people has at least three color balls in their pocket

Question

$50$ people in total, $30$ of them have a red ball, $36$ of them have a yellow ball, $44$ of them have a blue ball. Question: at least how many people has three balls?

Answer 1

Let $A$ be the set of people having a red ball, $B$ the set of people having a yellow ball and $C$ the set of people having a blue ball. Then we're after $\mid A \cap B \cap C \mid$. This is a classic Inclusion-exclusion principle question. Using this we have:

$$\mid \;A \cap B \cap C \mid \; = \; \mid A \cup B \cup C \mid - \mid A \mid - \mid B \mid - \mid C \mid + \mid A \cap B \mid + \mid B \cap C \mid + \mid C \cap A \;\mid$$

Now use the Pigeonhole Principle to find the minimum values of $\mid \; A \cap B \mid, \mid B \cap C \mid, \mid C \cap A \; \mid$