People Voting Combinations

In a group of 15 people, 12 voted for pizza, 10 for sushi, and 7 for tacos. If some voted for two or three types, calculate the number of people who voted for each combination (voted once, twice, or all three).

This problem dives into the concepts of set theory and combinatorics, specifically focusing on the principles of inclusion-exclusion. When solving problems that involve overlapping categories, such as this one where people could vote for multiple options, it is crucial to think about how different combinations of these categories intersect. By understanding the overlaps and unique counts, one can determine how many people belong to each combination of categories.

First, consider the principle of inclusion-exclusion which helps in counting the number of elements of the union of several sets. In simple terms, it accounts for adding the sizes of individual sets, then subtracting the sizes of all the pairwise intersections because they get added twice, and finally adding back the intersection of all three sets if it's the case of three sets. Applying these principles can aid in deducing accurate counts for each possible voting combination.

By solving this problem, students can enhance their understanding of how to apply set operations and the inclusion-exclusion principle in real-life scenarios. It broadens their capacity to think abstractly about how elements in groups relate, which is a key skill in both mathematical reasoning and computer science applications. This exercise not only strengthens conceptual knowledge but also improves problem-solving strategies often utilized in data analysis and algorithm design.