There are a total of 152 foreign language students in a high school where they offer only Spanish, French, and German.

You are given the following details for this semester:

58 take Spanish.

71 take French.

66 take German.

9 take Spanish and French but not German.

9 take Spanish and German but not French.

15 take French and German but not Spanish.

38 students are taking at least two languages.

1. How many students take all three languages in this semester?

2. How many students take only French in this semester?

3. What is the exact probability that two students selected from this school take French? Enter your answer as a rational number:

Use Venn diagrams.

Make 3 overlapping circles, and label them S, F, and G
Place x in the intersection of all 3
and then place the 9, 9, and 15 in their correct regions
In the "Spanish only" part, put in 40-x
In the "French only" part, put in 47-x
In the "German only" part, put in 42-x

We are told 38 are taking at least two languages, that would be the regions showing some overlap, so
9+9+15+x = 38
x = 5

1. 5 take all 3
2. 42 take only French
3. prob = (71/152)(70/151) = 2485/11476

To solve this problem, we can use a Venn diagram to represent the information given.

Let's label the three sets as:
S = Spanish
F = French
G = German

1. To find the number of students who take all three languages, we first need to find the number of students taking at least two languages. From the information given, we know that 38 students are taking at least two languages. Let's add this to our Venn diagram.

S⋂F = 9 (students taking Spanish and French but not German)
S⋂G = 9 (students taking Spanish and German but not French)
F⋂G = 15 (students taking French and German but not Spanish)
S⋂F⋂G = ? (students taking all three languages)

We can find S⋂F⋂G by subtracting the number of students taking at least two languages from the total number of students taking each language:
52 (students taking Spanish only) + 62 (students taking French only) + 57 (students taking German only) + 38 (students taking at least two languages) = 209

152 (total number of foreign language students) - 209 (students taking other combinations) = S⋂F⋂G

Therefore, the number of students taking all three languages is 152 - 209 = -57. However, this is not a possible value, so there may be an error in the given information.

2. To find the number of students taking only French in this semester, we need to subtract the number of students taking other languages or combinations from the total number of students taking French.

Students taking only French = 71 (students taking French) - (S⋂F + F⋂G) = 71 - (9 + 15) = 71 - 24 = 47.

Therefore, 47 students take only French in this semester.

3. To find the exact probability that two students selected from this school take French, we need to calculate the ratio of the number of students taking French to the total number of students.

Probability = Number of students taking French / Total number of students

Probability = 71 / 152.

The exact probability that two students selected from this school take French is 71/152.

To solve this problem, we can use a method called the principle of inclusion-exclusion. This principle allows us to calculate the number of students who fall into different categories by subtracting the overlapping counts.

1. To find the number of students taking all three languages, we need to find the intersection of all three sets: Spanish, French, and German. According to the given information, 38 students are taking at least two languages. However, we need to be careful as some of these students may be taking all three languages. To account for this, we need to subtract the number of students taking exactly two languages.

Let's calculate it step by step:

- Students taking Spanish and French: 9 (given)
- Students taking Spanish and German: 9 (given)
- Students taking French and German: 15 (given)

Now, let's subtract the students taking exactly two languages:

- (9 + 9 + 15) - 38 = 4

So, there are 4 students taking all three languages.

2. To find the number of students taking only French, we need to subtract the students taking other languages or combinations involving French from the total number of students taking French. Let's calculate it step by step:

- Students taking French and Spanish: 9 (given)
- Students taking French and German: 15 (given)
- Students taking all three languages: 4 (calculated in the previous step)

Now, let's subtract these counts from the total number of students taking French:

- 71 - (9 + 15 + 4) = 43

So, there are 43 students taking only French.

3. To find the exact probability that two students selected from this school take French, we need to consider the total number of possible pairs of students and the number of pairs that include at least one student taking French.

The total number of possible pairs can be calculated as the combination of 152 students taken 2 at a time:

- Total pairs = C(152, 2) = 152 * (152 - 1) / 2 = 152 * 151 / 2 = 22,876

Now, let's calculate the number of pairs that include at least one student taking French:

- Pairs with at least one student taking French = Total pairs - Pairs with no French

We already know that the number of students not taking French is 152 - 71 = 81. So, the number of pairs with no French is C(81, 2) = 81 * (81 - 1) / 2 = 3,240.

Therefore, the number of pairs with at least one student taking French is:

- Pairs with at least one student taking French = 22,876 - 3,240 = 19,636

The probability is the number of pairs with at least one student taking French divided by the total possible pairs:

- Probability = Pairs with at least one student taking French / Total pairs = 19,636 / 22,876

The exact probability that two students selected from this school take French is 19,636 / 22,876.