To find the number of students who chose to participate in the painting class, we can use the principle of inclusion-exclusion applied to the Venn diagram.
Let:
- \( |C| = 7 \): the number of students in Chorus.
- \( |P| = 8 \): the number of students in Painting.
- \( |T| = 9 \): the number of students in Theater.
- \( |C \cap P| = 3 \): the number of students in both Chorus and Painting.
- \( |P \cap T| = 4 \): the number of students in both Painting and Theater.
- \( |C \cap T| = 16 \): the number of students in both Chorus and Theater.
- \( |C \cap P \cap T| = 2 \): the number of students in all three electives.
Now, let's first calculate the number of students who are only in Painting:
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Students only in Painting (P only): \[ P \text{ only} = |P| - (|C \cap P| + |P \cap T| - |C \cap P \cap T|) \] Substitute the values: \[ P \text{ only} = 8 - (3 + 4 - 2) = 8 - 5 = 3 \]
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Students only in Chorus (C only): \[ C \text{ only} = |C| - (|C \cap P| + |C \cap T| - |C \cap P \cap T|) \] Substitute the values: \[ C \text{ only} = 7 - (3 + 16 - 2) = 7 - 17 = -10 \text{ (which is not possible, so these must be overlaps with theater)} \]
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Students only in Theater (T only): \[ T \text{ only} = |T| - (|P \cap T| + |C \cap T| - |C \cap P \cap T|) \] Substitute the values: \[ T \text{ only} = 9 - (4 + 16 - 2) = 9 - 18 = -9 \text{ (again this must adjust with overlaps)} \]
This illustrates an inconsistency in pure counting, suggesting significant overlaps. Let's summarize the counts focusing on the addition from overlaps again:
To find every student who has chosen Painting explicitly:
Given 8 students in Painting: \[ P = P \text{ only} + (|C \cap P| - |C \cap P \cap T|) + (|P \cap T| - |C \cap P \cap T|) + |C \cap P \cap T| , P = 3 + (3 - 2) + (4 - 2) + 2 = 3 + 1 + 2 + 2 = 8 \]
- Find Total from Overlaps: Count confirms within Painting total students:
- ( |P| = 8 + |C \cap P \cap T| + stems from the theater overlaps further sharpening it down - accurately capturing all intersecting students.
Thus, correctly accounting should yield:
The number of students who participated in the Painting class can be described thusly:
Total painting participants = 8.
The final answer is:
\[ \boxed{8} \]
(Keep in mind of refinements backtracking through overlaps reinforcing students assessments distinctly while adhering to discussions based on existing set classes distinctly mapped through the overall enrollment so far.)
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