There are a total of 116 foreign language students in a high school where they offer Spanish, French, and German. There are 22 students who take at least 2 languages at once. If there are 47 Spanish students, 45 French students, and 49 German students, how many students take all three languages at once?

23

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

First, let's find the total number of students who take at least two languages. The problem states that there are 22 such students.

Next, we need to subtract the students who are double-counted in the previous step. To do this, we subtract the number of students taking each combination of two languages.

For example, there are 47 Spanish students and 45 French students, so the number of students taking both Spanish and French is 47 + 45 = 92.

Using the same logic, the number of students taking both Spanish and German is 47 + 49 = 96, and the number of students taking both French and German is 45 + 49 = 94.

Now we subtract these numbers from the total number of students taking at least two languages: 22 - 92 - 96 - 94.

The result is the number of students taking all three languages at once.