A group of 123 workers went to a canteen for cold drinks, ice-cream and tea, 42 workers took ice-cream, 36 tea and 30 cold drinks. 15 workers purchased ice-cream and tea, 10 ice-cream and cold drinks, and 4 cold drinks and tea but not ice-cream, 11 took ice-cream and tea but not cold drinks. Determine how many workers did not purchase anything?

Explanation

To solve the problem step by step, we will use the principle of inclusion-exclusion to find out how many workers did not purchase anything.

### Step 1: Define the Sets
Let:
- \( I \) = set of workers who took ice cream
- \( T \) = set of workers who took tea
- \( C \) = set of workers who took cold drinks

Given:
- Total workers \( |U| = 123 \)
- \( |I| = 42 \) (workers who took ice cream)
- \( |T| = 36 \) (workers who took tea)
- \( |C| = 30 \) (workers who took cold drinks)

### Step 2: Define the Intersections
From the problem, we have:
- \( |I \cap T| = 15 \) (workers who took both ice cream and tea)
- \( |I \cap C| = 10 \) (workers who took both ice cream and cold drinks)
- \( |C \cap T| = 4 \) (workers who took cold drinks and tea but not ice cream)
- \( |I \cap T \cap C'| = 11 \) (workers who took ice cream and tea but not cold drinks)

### Step 3: Calculate the Intersections
From the information given, we can derive:
1. Workers who took all three items \( |I \cap T \cap C| \):
\[ |I \cap T| - |I \cap T \cap C' = 15 - 11 = 4 \]
So, \( |I \cap T \cap C| = 4 \).

2. Workers who took cold drinks and tea \( |C \cap T| \):
\[ |C \cap T| = 4 \quad (\text{given, but this includes only those who took cold drinks and tea without ice cream}) \]
We need to find \( |C \cap T| \) including those who took ice cream:
\[ |C \cap T| = |C \cap T| + |I \cap T \cap C| = 4 + 4 = 8 \]

### Step 4: Use Inclusion-Exclusion Principle
Now, we can find the total number of workers who purchased at least one item:
\[ |I \cup T \cup C| = |I| + |T| + |C| - |I \cap T| - |I \cap C| - |C \cap T| + |I \cap T \cap C| \]
Substituting the values:
\[ |I \cup T \cup C| = 42 + 36 + 30 - 15 - 10 - 8 + 4 \]
Calculating:
\[ |I \cup T \cup C| = 42 + 36 + 30 = 108 \]
\[ 108 - 15 - 10 - 8 + 4 = 108 - 33 + 4 = 79 \]

### Step 5: Calculate Workers Who Did Not Purchase Anything
To find the number of workers who did not purchase anything:
\[ \text{Workers who did not purchase anything} = |U| - |I \cup T \cup C| \]
Substituting the values:
\[ \text{Workers who did not purchase anything} = 123 - 79 = 44 \]

### Final Answer
Therefore, the number of workers who did not purchase anything is **44**.

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