1. **State the problem:** We have 18 students in the Math Club, 22 students in the Science Club, and 10 students in both clubs. We want to find: - Number of students only in Math Club - Number of students only in Science Club - Number of students in neither club 2. **Use the principle of inclusion-exclusion:** The total number of students in either club is given by: $$|M \cup S| = |M| + |S| - |M \cap S|$$ where $|M|=18$, $|S|=22$, and $|M \cap S|=10$. 3. **Calculate the number of students in either club:** $$|M \cup S| = 18 + 22 - 10 = 30$$ 4. **Find the number of students only in Math Club:** Students only in Math Club are those in Math but not in Science: $$|M \text{ only}| = |M| - |M \cap S| = 18 - 10 = 8$$ 5. **Find the number of students only in Science Club:** Students only in Science Club are those in Science but not in Math: $$|S \text{ only}| = |S| - |M \cap S| = 22 - 10 = 12$$ 6. **Find the number of students in neither club:** Assuming the total number of students is the union plus those in neither, let total students be $T$. We are not given $T$, so we cannot find the exact number in neither club without it. If we assume the total number of students is the sum of all unique students in clubs plus those in neither, then: $$T = |M \cup S| + |\text{neither}|$$ Since $T$ is not given, the number in neither club cannot be determined from the information provided. **Final answers:** - Only Math Club: $8$ - Only Science Club: $12$ - Neither club: Cannot be determined without total number of students