1. **State the problem:** We need to find how many students are on the Awards committee or the Prom committee. 2. **Identify the sets and values from the Venn diagram:** - Only Awards: 6 - Only Prom: 4 - Field Day $\cap$ Prom: 5 - Field Day $\cap$ Awards: 2 - All three committees: 10 - Outside all committees: 9 3. **Recall the formula for union of two sets:** $$|A \cup P| = |A| + |P| - |A \cap P|$$ where $A$ is Awards committee, $P$ is Prom committee. 4. **Calculate $|A|$ and $|P|$:** - $|A| = \text{Only Awards} + \text{Field Day} \cap \text{Awards} + \text{All three} = 6 + 2 + 10 = 18$ - $|P| = \text{Only Prom} + \text{Field Day} \cap \text{Prom} + \text{All three} = 4 + 5 + 10 = 19$ 5. **Calculate $|A \cap P|$:** - $|A \cap P| = \text{All three} = 10$ (since the only intersection between Awards and Prom is through all three committees) 6. **Apply the union formula:** $$|A \cup P| = 18 + 19 - 10 = 27$$ 7. **Answer:** There are **27** students on the Awards committee or the Prom committee.