The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine the foreign language courses they had taken while undergraduates at the college. Of the 500 graduate

207 had at least one year of Spanish.
171 had at least one year of French.
140 had at least one year of German.
40 had at least one year of Spanish and French.
25 had at least one year of Spanish and German.
22 had at least one year of French and German.
4 had at least one year of all three languages.

(a) How many of the graduates had at least 1 yr of at least one of the three languages?
(b) How many of the graduates had at least 1 yr of exactly one of the three languages?
(c) How many of the graduates had less than 1 yr of any of the three languages?

This is a typical 3-circle Venn diagram. When you fill in the areas, the question should be easy.

Start with the central 3-circle intersection. That is 4.

...

To solve this problem, we can use a principle called the inclusion-exclusion principle. This principle allows us to calculate the number of elements in the union of multiple sets by considering the number of elements in each individual set, as well as the intersections between the sets.

Let's break down the given information into a Venn diagram to visualize the relationships between the sets:

```
S (Spanish) F (French) G (German)
207 171 140
---+---------+-----------+------------
| | |
40| | | x
| | |
| +-----+-----+
| | x |
| | |
+----+----+-----------+---
| x | |
| | |
| | |
| | |
```

Now, let's use this information to answer each part of the question:

(a) How many of the graduates had at least 1 year of at least one of the three languages?

To calculate this, we need to sum up the number of graduates who had at least 1 year of each language and subtract the number of graduates who had at least 1 year of the intersections between the languages. However, some graduates are counted multiple times in these intersections, so we need to correct for that.

Let's calculate it step by step:

- Number of graduates who had at least 1 year of Spanish (S): 207
- Number of graduates who had at least 1 year of French (F): 171
- Number of graduates who had at least 1 year of German (G): 140

Intersections:

- Number of graduates who had at least 1 year of both Spanish and French (S ∩ F): 40
- Number of graduates who had at least 1 year of both Spanish and German (S ∩ G): 25
- Number of graduates who had at least 1 year of both French and German (F ∩ G): 22

Intersection of all three languages (S ∩ F ∩ G): 4

To calculate the total number of graduates who had at least 1 year of at least one of the three languages (S ∪ F ∪ G), we can use the inclusion-exclusion principle:

(S ∪ F ∪ G) = (S + F + G) - (S ∩ F + S ∩ G + F ∩ G) + (S ∩ F ∩ G)

(S ∪ F ∪ G) = (207 + 171 + 140) - (40 + 25 + 22) + 4

Now, calculate (S ∪ F ∪ G) to get the answer.

(b) How many of the graduates had at least 1 year of exactly one of the three languages?

To calculate this, we need to consider the sets that have only one language, without any intersections. These are S - (S ∩ F ∪ S ∩ G), F - (S ∩ F ∪ F ∩ G), and G - (S ∩ G ∪ F ∩ G). Calculate the sum of these sets to get the answer.

(c) How many of the graduates had less than 1 year of any of the three languages?

To calculate this, we subtract the number of graduates who had at least 1 year of at least one of the three languages from the total number of graduates (500). Calculate the difference to get the answer.

I hope this helps you understand how to approach this problem! Let me know if you have any further questions.