To solve this question, we will use the principle of inclusion and exclusion. Let's break down the problem step by step.
We are given the following information:
Total students: 100
10 students had all three drinks (milk, coffee, and tea).
20 students had milk and coffee.
30 students had coffee and tea.
25 students had milk and tea.
12 students had only milk.
5 students had only coffee.
8 students had only tea.
We are asked to find the number of students who did not take any of the three drinks.
Let’s define:
M = number of students who drink milk.
C = number of students who drink coffee.
T = number of students who drink tea.
Using the given data:
|M ∩ C ∩ T| = 10 (students who had all three drinks)
|M ∩ C| = 20 (students who had milk and coffee)
|C ∩ T| = 30 (students who had coffee and tea)
|M ∩ T| = 25 (students who had milk and tea)
|M only| = 12 (students who had only milk)
|C only| = 5 (students who had only coffee)
|T only| = 8 (students who had only tea)
Step 1: Apply the principle of inclusion and exclusion
The formula for the number of students who had at least one of the drinks is:
|M ∪ C ∪ T| = |M| + |C| + |T| - |M ∩ C| - |C ∩ T| - |M ∩ T| + |M ∩ C ∩ T|
Where:
|M| is the total number of students who drink milk.
|C| is the total number of students who drink coffee.
|T| is the total number of students who drink tea.
|M ∩ C| is the number of students who drink both milk and coffee.
|C ∩ T| is the number of students who drink both coffee and tea.
|M ∩ T| is the number of students who drink both milk and tea.
|M ∩ C ∩ T| is the number of students who drink all three drinks.
Step 2: Find the number of students who had milk, coffee, or tea
First, let’s calculate the number of students in each of the different categories:
Students who had both milk and coffee but not tea = |M ∩ C| - |M ∩ C ∩ T| = 20 - 10 = 10
Students who had both coffee and tea but not milk = |C ∩ T| - |M ∩ C ∩ T| = 30 - 10 = 20
Students who had both milk and tea but not coffee = |M ∩ T| - |M ∩ C ∩ T| = 25 - 10 = 15
Now, let’s find the total number of students who drank at least one of the drinks:
|M ∪ C ∪ T| = (12 students who had only milk) + (5 students who had only coffee) + (8 students who had only tea) + (10 students who had milk and coffee but not tea) + (20 students who had coffee and tea but not milk) + (15 students who had milk and tea but not coffee) + (10 students who had all three drinks)
|M ∪ C ∪ T| = 12 + 5 + 8 + 10 + 20 + 15 + 10 = 80
Step 3: Calculate the number of students who did not take any drinks
Since there are 100 students in total and 80 students took at least one of the drinks, the number of students who did not take any of the three drinks is:
100 - 80 = 20
Final Answer:
The number of students who did not take any of the three drinks is 20.
