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 determine how many workers did not purchase anything, we can use the principle of inclusion-exclusion and Venn diagrams. Let's break down the problem step by step.

### Step 1: Define the Sets
Let:
- \( A \): the set of workers who took ice-cream.
- \( B \): the set of workers who took tea.
- \( C \): the set of workers who took cold drinks.

From the problem, we have:
- \( |A| = 42 \) (workers took ice-cream)
- \( |B| = 36 \) (workers took tea)
- \( |C| = 30 \) (workers took cold drinks)

### Step 2: Define the Intersections
We also know the following intersections:
- \( |A \cap B| = 15 \) (workers took both ice-cream and tea)
- \( |A \cap C| = 10 \) (workers took both ice-cream and cold drinks)
- \( |B \cap C| = 4 \) (workers took both tea and cold drinks, but not ice-cream)
- \( |A \cap B| - |A \cap B \cap C| = 11 \) (workers took ice-cream and tea but not cold drinks)

Let \( x = |A \cap B \cap C| \) (workers who took all three).

### Step 3: Set Up Equations
From the information given:
1. From \( |A \cap B| = 15 \):
\[ |A \cap B| = |A \cap B| - |A \cap B \cap C| + |A \cap B \cap C| \implies 15 = 11 + x \implies x = 4 \]

2. From \( |A \cap C| = 10 \):
\[ |A \cap C| = |A \cap C| - |A \cap B \cap C| + |A \cap B \cap C| \implies 10 = |A \cap C| - x \implies |A \cap C| = 10 - 4 = 6 \]

3. From \( |B \cap C| = 4 \):
\[ |B \cap C| = 4 \text{ (given that these workers did not take ice-cream)} \]

### Step 4: Calculate Total Workers Who Purchased
Using the principle of inclusion-exclusion:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]
Substituting the values:
\[ |A \cup B \cup C| = 42 + 36 + 30 - 15 - 10 - 4 + 4 \]
Calculating this gives:
\[ |A \cup B \cup C| = 42 + 36 + 30 - 15 - 10 - 4 + 4 = 83 \]

### Step 5: Calculate Workers Who Did Not Purchase Anything
The total number of workers is 123. Therefore, the number of workers who did not purchase anything is:
\[ \text{Workers who did not purchase} = \text{Total workers} - |A \cup B \cup C| = 123 - 83 = 40 \]

### Final Answer
Thus, the number of workers who did not purchase anything is **40**.

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