
In a student campus having 420 residents, everybody speaks at least one of the three languages
- French, English and Spanish. The number of people who speak French is 200, the number of
people who speak English is 300 and the number of people who speak Spanish is 400. What is
the minimum possible number of students who speak all the three languages?

To find the minimum possible number of students who speak all three languages, we can use the principle of inclusion-exclusion (PIE).

Let's denote:
- \( F \) = number of students who speak French = 200
- \( E \) = number of students who speak English = 300
- \( S \) = number of students who speak Spanish = 400
- \( n \) = number of students who speak all three languages = ?

According to the principle of inclusion-exclusion, the total number of students who speak at least one of the three languages is given by:

\[
N(F \cup E \cup S) = F + E + S - (F \cap E) - (E \cap S) - (S \cap F) + (F \cap E \cap S)
\]

Given that \( N(F \cup E \cup S) = 420 \) (since everyone speaks at least one language), we have:

\[
420 = 200 + 300 + 400 - (F \cap E) - (E \cap S) - (S \cap F) + n
\]

Simplifying this:

\[
420 = 900 - ((F \cap E) + (E \cap S) + (S \cap F)) + n
\]

\[
(F \cap E) + (E \cap S) + (S \cap F) = 900 + n - 420
\]

\[
(F \cap E) + (E \cap S) + (S \cap F) = 480 + n
\]

To minimize \( n \), we need to maximize the sum of \( (F \cap E) + (E \cap S) + (S \cap F) \). Since each intersection (e.g., \( F \cap E \)) involves at least two languages, the maximum possible value for each pairwise intersection will be when it reaches the number of students speaking the less frequent language.

To calculate \( (F \cap E) + (E \cap S) + (S \cap F) \) under the condition that \( n \) is minimized, note:

\[
(F \cap E \cap S) \text{ is non-negative and occurs when pairwise overlaps are maximized.}
\]

The maximum occurs when the pairwise overlaps sum exactly equals 480 (because all intersections involving exactly two languages can't exceed the smallest total sum for the union 900 students).

Thus, when considering only 900, you subtract 420. Therefore, the minimum possible value of \( n \) can be calculated directly:

\[
n = 900 - 480 - 420 = 0
]

This is impossible under our original constraints (because all must speak a language).

Given that it can't be zero, let's adjust and solve the equation to find the minimum number. Instead, it's best done by a direct calculation.

If the sum is \( F + E + S \) subtracting the largest sum, then intersection is exactly 480, you get the following minimum result:

Result: **80 students**

Thus, the minimum number of students who must speak all three languages is **80**.

Would you like to clarify or review?

Calculate the minimum number of students who speak French, English, and Spanish simultaneously using the principle of inclusion-exclusion. Understand the intersections and union of language speakers among 420 residents on a student campus.

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