Three popular courses at Oliver Senior HS are Orchestra, Probability/Statistic, and Art. A survey to 200 students showed the following:
70 take Orchestra
80 take Probability/Statistics
60 take Art
35 take Orchestra and Probability/Statistic
33 take Orchestra and Art
31 take Probability/Statistics and Art
15 take all three
How many students had none of these courses?

To find how many students at Oliver Senior HS take none of the courses, we can use the principle of inclusion-exclusion. 

Here is a breakdown of how we proceed:

### Let:
- \( O \) represent the set of students taking Orchestra.
- \( P \) represent the set of students taking Probability/Statistics.
- \( A \) represent the set of students taking Art.

We are given:
- \( |O| = 70 \) (students taking Orchestra)
- \( |P| = 80 \) (students taking Probability/Statistics)
- \( |A| = 60 \) (students taking Art)
- \( |O \cap P| = 35 \) (students taking both Orchestra and Probability/Statistics)
- \( |O \cap A| = 33 \) (students taking both Orchestra and Art)
- \( |P \cap A| = 31 \) (students taking both Probability/Statistics and Art)
- \( |O \cap P \cap A| = 15 \) (students taking all three courses)

### Step 1: Use inclusion-exclusion to find how many students take at least one course.
The formula for inclusion-exclusion is:

\[
|O \cup P \cup A| = |O| + |P| + |A| - |O \cap P| - |O \cap A| - |P \cap A| + |O \cap P \cap A|
\]

Substitute the given values:

\[
|O \cup P \cup A| = 70 + 80 + 60 - 35 - 33 - 31 + 15
\]

Now, let's calculate it:

\[
|O \cup P \cup A| = 210 - 99 + 15 = 126
\]

So, 126 students take at least one of the three courses.

### Step 2: Find the number of students who take none of the courses.
The total number of students surveyed is 200. Therefore, the number of students who take none of the courses is:

\[
\text{None} = 200 - |O \cup P \cup A| = 200 - 126 = 74
\]

### Final Answer:
The number of students who take none of the courses is **74**.

Do you want more details or have any questions?

Here are some related questions you might consider:
1. How many students take exactly one course?
2. How many students take exactly two courses?
3. What percentage of students take Orchestra?
4. How many students take either Orchestra or Probability/Statistics, but not both?
5. What is the probability that a randomly selected student takes Art?

**Tip:** The principle of inclusion-exclusion is very useful when solving problems involving overlapping sets.

A survey of 200 students at Oliver Senior HS showed that 70 take Orchestra, 80 take Probability/Statistics, 60 take Art, with overlaps between these groups. Using the inclusion-exclusion principle, find out how many students take none of the courses. Ideal for high school students (Grades 9-12) learning set theory.

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