To solve the problem of how many market women are in the group, we can use the principle of inclusion-exclusion. This principle helps us count the total number of elements in overlapping sets without double-counting those that belong to multiple sets. ### Step-by-Step Explanation 1. **Define the Sets**: - Let \( M \) be the set of women who sell maize. - Let \( Y \) be the set of women who sell yam. - Let \( P \) be the set of women who sell plantain. From the problem, we have: - \( |M| = 12 \) (women selling maize) - \( |Y| = 10 \) (women selling yam) - \( |P| = 14 \) (women selling plantain) 2. **Define the Intersections**: - \( |M \cap P| = 5 \) (women selling both maize and plantain) - \( |M \cap Y| = 4 \) (women selling both maize and yam) - \( |Y \cap P| = 2 \) (women selling both yam and plantain only) - \( |M \cap Y \cap P| = 3 \) (women selling all three items) 3. **Use Inclusion-Exclusion Principle**: The formula for the total number of women in the group, denoted as \( |M \cup Y \cup P| \), is given by: \[ |M \cup Y \cup P| = |M| + |Y| + |P| - |M \cap Y| - |M \cap P| - |Y \cap P| + |M \cap Y \cap P| \] 4. **Substituting the Values**: Now, we substitute the values we have into the formula: \[ |M \cup Y \cup P| = 12 + 10 + 14 - 4 - 5 - 2 + 3 \] 5. **Calculating Step-by-Step**: - First, add the individual sets: \[ 12 + 10 + 14 = 36 \] - Next, subtract the intersections: \[ 36 - 4 - 5 - 2 = 36 - 11 = 25 \] - Finally, add back the women who sell all three items: \[ 25 + 3 = 28 \] 6. **Final Count**: Thus, the total number of women in the group is: \[ |M \cup Y \cup P| = 28 \] ### Analyzing the Options Now, let's compare our calculated total with the provided options: - A. 25 - B. 19 - C. 18 - D. 17 None of the options match our calculated total of 28. Therefore, it seems there may have been a misunderstanding in the interpretation of the problem or the options provided. ### Common Pitfalls - **Double Counting**: When counting women who sell multiple items, it’s crucial to subtract the overlaps correctly to avoid double counting. - **Misinterpretation of Intersections**: Ensure that the intersections are understood correctly, especially when it comes to those selling only two items versus all three. ### Revision Summary - Use the inclusion-exclusion principle to count elements in overlapping sets. - Carefully define the sets and their intersections. - Substitute values into the inclusion-exclusion formula step-by-step. - Always check your calculations and ensure you understand the problem context. In conclusion, based on the calculations, the total number of women in the group is 28, which does not match any of the provided options. It’s important to verify the problem statement and the options given.