There are a total of 103 foreign language students in a high school where they offer Spanish, French, and German. There are 29 students who take at least 2 languages at once. If there are 40 Spanish students, 42 French students, and 46 German students, how many students take all three languages at once?

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h.j.o;k

To find the number of students who take all three languages at once, we need to determine the intersection of the Spanish, French, and German students.

We are given that there are a total of 103 foreign language students in the high school. We are also given that there are 29 students who take at least 2 languages at once.

Let's break down the information we have:

- The number of Spanish students is 40.
- The number of French students is 42.
- The number of German students is 46.
- The total number of foreign language students is 103.
- The number of students who take at least 2 languages is 29.

To find the number of students who take all three languages at once, we can use the principle of inclusion-exclusion.

Step 1: Find the total number of students who take at least two languages:
This is already given as 29.

Step 2: Find the total number of students who take two languages:
We need to subtract the number of students who take all three languages from the total number of students who take at least two languages. Let's represent the number of students who take all three languages as 'x'.

29 = x + number of students who take exactly two languages.

Step 3: Find the number of students who take exactly two languages:
To find this, we need to add the number of students who take Spanish and French but not German, Spanish and German but not French, and French and German but not Spanish.

Let's use A, B, and C to represent the number of students who take Spanish and French but not German, Spanish and German but not French, and French and German but not Spanish, respectively.

We know that the total number of students who take exactly two languages is equal to A + B + C.

Step 4: Find the number of students who take all three languages (x):
Since the total number of foreign language students is 103, we can subtract the number of students who take exactly two languages (A + B + C) from this total.

103 = x + (A + B + C).

We have three unknowns (A, B, C) and two equations, both of which involve the same unknowns, so we can't solve for any specific values.

Therefore, without additional information about the value of A, B, or C, we can't determine the exact number of students who take all three languages.