Let's define the sets:

• Let $S$ be the set of students who take Spanish.
• Let $F$ be the set of students who take French.
• Let $G$ be the set of students who take German.

The given information is:

• $|S| = 50$ (students who take Spanish),
• $|F| = 42$ (students who take French),
• $|G| = 50$ (students who take German),
• $|S \cup F \cup G| = 117$ (total number of foreign language students),
• 29 students take at least 2 languages.

We are tasked with finding the number of students who take all three languages, which we'll denote as $x$.

Step 1: Apply the Principle of Inclusion-Exclusion

The principle of inclusion-exclusion for three sets is given by the formula:

$|S \cup F \cup G| = |S| + |F| + |G| - |S \cap F| - |F \cap G| - |G \cap S| + |S \cap F \cap G|$

Here:

• $|S \cap F \cap G| = x$ is the number of students taking all three languages.
• The sum of students taking at least 2 languages is given as 29.

We need to express the total number of students taking exactly two languages in terms of $x$. Define:

• $|S \cap F|$ as the number of students taking both Spanish and French,
• $|F \cap G|$ as the number of students taking both French and German,
• $|G \cap S|$ as the number of students taking both German and Spanish.

Let $a = |S \cap F| - x$, $b = |F \cap G| - x$, and $c = |G \cap S| - x$, where $a + b + c + 3x = 29$. The