1. **State the problem:** We need to find the number of students who are on the Prom and Awards Committees, or on the Field Day Committee. 2. **Identify the sets and values from the Venn diagram:** - Prom only: 4 - Awards only: 6 - Field Day only: 8 - Prom $\cap$ Awards only (excluding Field Day): 3 - Prom $\cap$ Field Day only (excluding Awards): 5 - Awards $\cap$ Field Day only (excluding Prom): 2 - Prom $\cap$ Awards $\cap$ Field Day: 10 - Outside all committees: 9 3. **Define the sets:** Let - $P$ = Prom Committee - $A$ = Awards Committee - $F$ = Field Day Committee 4. **Express the problem in set notation:** We want to find $|(P \cap A) \cup F|$. 5. **Use the formula for union of two sets:** $$|(P \cap A) \cup F| = |P \cap A| + |F| - |(P \cap A) \cap F|$$ 6. **Calculate each term:** - $|P \cap A|$ includes those in both Prom and Awards, which are the regions with 3 (Prom and Awards only) and 10 (all three committees), so $$|P \cap A| = 3 + 10 = 13$$ - $|F|$ includes all in Field Day, which are 8 (Field Day only), 5 (Prom and Field Day only), 2 (Awards and Field Day only), and 10 (all three), so $$|F| = 8 + 5 + 2 + 10 = 25$$ - $|(P \cap A) \cap F|$ is the number in all three committees, which is 10. 7. **Substitute values into the formula:** $$|(P \cap A) \cup F| = 13 + 25 - 10 = 28$$ 8. **Final answer:** There are **28** students who are on the Prom and Awards Committees, or on the Field Day Committee.