HOW 480 COMES IN PIC ??? PLEASE BROTHER PROVIDE A CLEAR SOLUTION ONLY FOR LAST PART

To understand where the 480 comes from, we need to focus on the goal of this problem: maximizing the students who study nothing.

To maximize those who study nothing, we must minimize the total number of students who study at least one language. This is achieved by "hiding" the Spanish (S) and German (G) students inside the French (F) circle as much as possible.

1. Breaking Down the French Circle (F = 1320)

The French circle is already fixed at 1320 students. We are told:

• Both French and Spanish ($F \cap S$): 408 students.
• All three ($F \cap S \cap G$): 108 students.
• Only French and Spanish: $408 - 108 = \mathbf{300}$ (This is the "300" seen in the diagram).

2. Where the "480" Comes From

The number 480 represents students who study French and German only. Here is the step-by-step logic:

• The Constraint: We want to minimize the total union. To do this, we should try to make sure nobodystudies only Spanish or only German. Every S and G student should ideally be inside the F circle.
• The Calculation: Total French students = 1320.
• Students already accounted for in F: 300 (F & S) + 108 (All three) + 432 (Only French) = 840.
• Remaining capacity in the French circle: $1320 - 840 = \mathbf{480}$.

By putting 480 students in the "French and German only" section, we have fully utilized the 1320 students in the French circle without adding any "new" students to the total count.

3. Final Logic for the Solution

The solution concludes that the maximum number of students who study nothing is:

$\text{Total Students} - \text{Total French Students} = 2116 - 1320 = \mathbf{796}$

This is possible only if all Spanish and German students fit inside that 1320-student French circle. The "480" is just a way to fill the Venn diagram to prove that such a distribution is mathematically possible.