In a group of 120 people: 65 eat Rice, 45 eat bread, 42 eat curd, 20 eat both Rice and bread, 25 eat both Rice and curd, 15 eat both bread and curd and 8 eat all three items. Which of the following is the number of people who eat at least one of the three items?

Correct option is B

To solve this problem, we will use the principle of Inclusion-Exclusion.
Given:
· Total people = 120
· People who eat Rice = 65
· People who eat Bread = 45
· People who eat Curd = 42
· People who eat both Rice and Bread = 20
· People who eat both Rice and Curd = 25
· People who eat both Bread and Curd = 15
· People who eat all three items = 8
We need to find the number of people who eat at least one of the three items (Rice, Bread or Curd).
Step 1: Apply the Inclusion-Exclusion Principle
The formula for the union of three sets, using Inclusion-Exclusion, is:
​$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$$​​
Where:
· A is the set of people who eat Rice
· B is the set of people who eat Bread
· C is the set of people who eat Curd
Step 2: Substitute the given values into the formula
· |A| = 65 (People who eat Rice)
· |B| = 45 (People who eat Bread)
· |C| = 42 (People who eat Curd)
· $$|A \cap B|$$ = 20 (People who eat both Rice and Bread)
· $$|B \cap C|$$​ = 15 (People who eat both Bread and Curd)
· $$|C \cap A|$$​ = 25 (People who eat both Rice and Curd)
· $$|A \cap B \cap C|$$​ = 8 (People who eat all three items)
Substitute into the formula:
$$|A \cup B \cup C|$$​ = 65 + 45 + 42 - 20 - 15 - 25 + 8
Step 3: Simplify the calculation
$$|A \cup B \cup C|$$​ = 152 - 60 + 8 = 100
Step 4: Conclusion
The number of people who eat at least one of the three items (Rice, Bread or Curd) is 100.
Information Booster:
1. Inclusion-Exclusion Principle: The Inclusion-Exclusion Principle is a way to count the number of elements in the union of several sets, especially when there is overlap between the sets. It works by adding the sizes of the individual sets, then subtracting the sizes of their pairwise intersections to avoid double-counting, and then adding the size of the intersection of all sets to correct for triple-counting.
2. Understanding the problem: This is a set theory problem where we are dealing with the union of three sets, which represent the people who eat Rice, Bread, and Curd. The principle is applied here to find the total number of people who eat at least one of the three items.
3. Real-world applications: The Inclusion-Exclusion Principle is widely used in probability theory, computer science (for database query optimization), and statistics to handle overlapping sets, ensuring that we count each element only once.
Additional Knowledge:
· Why the Inclusion-Exclusion Principle Works?
The principle works by adjusting for overcounts. When we simply add the sizes of the sets, we double-count the elements that appear in two sets. The second step subtracts these double-counted elements, but some elements might still be subtracted too much if they appear in all three sets. So, the final step adds back the triple intersection to get the accurate count.
· Other Set Operations: Apart from Inclusion-Exclusion, Venn diagrams are another useful tool in understanding and visualizing problems involving multiple sets. They show the relationships between different sets and make it easier to apply operations like union, intersection and difference.