I have added comments to your solution of the inclusion/exclusion problem at the end of the original post:
http://www.jiskha.com/display.cgi?id=1302620150
Feel free to go back and check it out.
As an AI text-based bot, I don't have the capability to access or browse external websites or specific posts. However, I can explain the inclusion/exclusion principle and help you understand how to approach such problems.
The inclusion/exclusion principle is a fundamental counting technique used to calculate the size of sets. It is commonly used in combinatorics and probability theory. The principle allows us to determine the size of the union of multiple sets by considering the sizes of individual sets and their intersections.
To apply the inclusion/exclusion principle, follow these steps:
1. Identify the sets involved: Determine the number of sets you are working with and list them.
2. Calculate the size of individual sets: Determine the number of elements in each set.
3. Calculate the size of pairwise intersections: Determine the number of elements shared between any two sets.
4. Calculate the size of three-way intersections and so on: Continue calculating the sizes of intersections involving three or more sets.
5. Apply the inclusion/exclusion principle formula: Use the formula:
|A U B U C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
Here, |X| represents the size of set X, and ∩ represents the intersection between sets.
6. Simplify the formula: Simplify the formula based on the given information. If certain intersections are empty or if sets are disjoint (have no common elements), you can exclude them from the formula.
By following these steps, you can apply the inclusion/exclusion principle to various counting problems involving sets and intersections.