Of the 40 eighth-graders at Pascal Middle School, 17 take 21. algebra, 109 take advanced computer, 14 take industrial technology. Furthermore, 70 take both algebra and advanced computer, 34 take both algebra and industrial technology, and
29 take both advanced computer and industrial technology. Finally, 164 students take none of these courses. How many
students take all three courses?

Let's define the following:

- \( A \) = set of students taking Algebra
- \( C \) = set of students taking Advanced Computer
- \( T \) = set of students taking Industrial Technology

### Given information:

- Total students at Pascal Middle School: \( 40 \)
- Students taking Algebra: \( |A| = 21 \)
- Students taking Advanced Computer: \( |C| = 109 \)
- Students taking Industrial Technology: \( |T| = 14 \)
- Students taking both Algebra and Advanced Computer: \( |A \cap C| = 70 \)
- Students taking both Algebra and Industrial Technology: \( |A \cap T| = 34 \)
- Students taking both Advanced Computer and Industrial Technology: \( |C \cap T| = 29 \)
- Students taking none of the three courses: \( 164 \)

We need to find the number of students who take all three courses, denoted as \( x \).

### Step 1: Apply the Principle of Inclusion-Exclusion
The formula for the union of three sets is:

\[
|A \cup C \cup T| = |A| + |C| + |T| - |A \cap C| - |A \cap T| - |C \cap T| + |A \cap C \cap T|
\]

We know that the number of students who take at least one of these courses is:

\[
40 - 164 = -124
\]

Now, plug in the known values:

\[
|A \cup C \cup T| = 21 + 109 + 14 - 70 - 34 - 29 + x
\]

Simplify the equation:

\[
|A \cup C \cup T| = 144 - 133 + x
\]

\[
|A \cup C \cup T| = 11 + x
\]

Since the total number of students who take at least one course is \( -124 \), we have:

\[
11 + x = -124
\]

Solving for \( x \):

\[
x = -124 - 11
\]
\[
x = -135
\]

Thus, **the number of students who take all three courses is \(-135\)**. However, this value doesn't

Of the 40 eighth-graders at Pascal Middle School, 17 take 21. algebra, 109 take advanced computer, 14 take industrial technology. Furthermore, 70 take both algebra and advanced computer, 34 take both algebra and industrial technology, and 29 take both advanced computer and industrial technology. Finally, 164 students take none of these courses. How many students take all three courses?

Learn how to solve a real-world problem involving set theory and the principle of inclusion-exclusion. Find how many students take all three courses: Algebra, Advanced Computer, and Industrial Technology, with detailed calculations. Suitable for Grades 8-10.

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