There are 600 boys in a hostel. Each plays either hockey or football or both. If 75% plays hockey and 45% play football, How many boys plays both ?

Options

A. a) 48
B. b) 60
C. c) 80
D. d) 120

Answer

To solve the problem step by step, we will use the information provided and apply the principle of set theory.

### Step 1: Determine the number of boys who play hockey
We know that 75% of the 600 boys play hockey.

\[ \text{Number of boys playing hockey} (N_A) = 75\% \text{ of } 600 = \frac{75}{100} \times 600 = 450 \]

### Step 2: Determine the number of boys who play football
We also know that 45% of the 600 boys play football.

\[ \text{Number of boys playing football} (N_B) = 45\% \text{ of } 600 = \frac{45}{100} \times 600 = 270 \]

### Step 3: Use the principle of inclusion-exclusion
We need to find the number of boys who play both hockey and football (denoted as \(N_{A \cap B}\)). We can use the formula for the union of two sets:

\[ N_{A \cup B} = N_A + N_B - N_{A \cap B} \]

Given that the total number of boys is 600, we can set up the equation:

\[ 600 = 450 + 270 - N_{A \cap B} \]

### Step 4: Solve for \(N_{A \cap B}\)
Now, we can rearrange the equation to solve for \(N_{A \cap B}\):

\[ 600 = 720 - N_{A \cap B} \]

\[ N_{A \cap B} = 720 - 600 \]

\[ N_{A \cap B} = 120 \]

### Conclusion
Thus, the number of boys who play both hockey and football is **120**.

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