In a class of 45 students each one plays either Cricket or Hockey. If 30 play Cricket and 27 play Hockey then :
how many play Hockey only ?

Answer

To solve the problem step by step, we will use the information provided in the question and apply set theory concepts.

### Step 1: Understand the problem
We have a total of 45 students in a class, and each student plays either Cricket or Hockey. We know:
- Number of students playing Cricket (Set A) = 30
- Number of students playing Hockey (Set B) = 27

### Step 2: Define the sets
Let:
- \( |A| \) = number of students who play Cricket = 30
- \( |B| \) = number of students who play Hockey = 27
- \( |A \cup B| \) = total number of students = 45
- \( |A \cap B| \) = number of students who play both Cricket and Hockey

### Step 3: Use the formula for union of sets
We can use the formula for the union of two sets:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Substituting the known values:
\[ 45 = 30 + 27 - |A \cap B| \]

### Step 4: Solve for \( |A \cap B| \)
Now, simplify the equation:
\[ 45 = 57 - |A \cap B| \]
Rearranging gives:
\[ |A \cap B| = 57 - 45 \]
\[ |A \cap B| = 12 \]

### Step 5: Find the number of students who play Hockey only
To find the number of students who play Hockey only, we subtract the number of students who play both sports from the total number of students who play Hockey:
\[ \text{Hockey only} = |B| - |A \cap B| \]
Substituting the values:
\[ \text{Hockey only} = 27 - 12 \]
\[ \text{Hockey only} = 15 \]

### Final Answer
The number of students who play Hockey only is **15**.

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