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:
Set Theory: Calculating People Eating At Least One Item
This problem requires us to find the total number of people who consume at least one of the three specified items: Rice, Bread, or Curd. We are given information about the total number of people in the group, the number who consume each item individually, the number who consume pairs of items, and the number who consume all three items. The mathematical tool best suited for this type of problem is the Principle of Inclusion-Exclusion, a core concept in set theory used for counting elements in unions of sets.
Data Summary for People Eating Food Items
| Item/Group | Number of People |
|---|---|
| Total People | 120 |
| Eat Rice (R) | 65 |
| Eat Bread (B) | 45 |
| Eat Curd (C) | 42 |
| Eat Rice and Bread (R & B) | 20 |
| Eat Rice and Curd (R & C) | 25 |
| Eat Bread and Curd (B & C) | 15 |
| Eat all three (R & B & C) | 8 |
Applying the Principle of Inclusion-Exclusion
The Principle of Inclusion-Exclusion allows us to calculate the size of the union of multiple sets accurately. For three sets, let R represent the set of people who eat Rice, B represent those who eat Bread, and C represent those who eat Curd. The number of people who eat at least one of these items is given by the size of the union of these sets, denoted as $|R \cup B \cup C|$. The formula is:
$ |R \cup B \cup C| = |R| + |B| + |C| - (|R \cap B| + |R \cap C| + |B \cap C|) + |R \cap B \cap C| $
Here's a conceptual explanation of the formula:
- We start by adding the counts of people in each individual set ($|R|$, $|B|$, $|C|$).
- People who belong to the intersection of two sets (e.g., Rice and Bread, $|R \cap B|$) are counted twice in the initial sum. So, we subtract the counts of all pairwise intersections ($(R \cap B)$, $(R \cap C)$, $(B \cap C)$) to correct for this overcounting.
- After subtracting the pairwise intersections, the people who belong to the intersection of all three sets ($|R \cap B \cap C|$) have been added three times initially and subtracted three times in the second step. This means they haven't been counted at all. Therefore, we must add the count of the triple intersection back to ensure they are included exactly once.
Step-by-Step Calculation for People Eating At Least One Item
Let's apply the formula using the provided numbers:
- Sum of individuals eating each item: $ |R| + |B| + |C| = 65 + 45 + 42 $ $ = 152 $
- Sum of individuals eating pairs of items: $ |R \cap B| + |R \cap C| + |B \cap C| = 20 + 25 + 15 $ $ = 60 $
- Number of individuals eating all three items: $ |R \cap B \cap C| = 8 $
- Calculate the total number eating at least one item using the Inclusion-Exclusion formula: $ |R \cup B \cup C| = (Sum\ of\ individuals) - (Sum\ of\ pairs) + (Triple\ intersection) $ $ |R \cup B \cup C| = 152 - 60 + 8 $ $ |R \cup B \cup C| = 92 + 8 $ $ |R \cup B \cup C| = 100 $
Final Result
The calculation shows that 100 people in the group eat at least one of the three items: Rice, Bread, or Curd.
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