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 Cricket only ?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the sets
Let:
- \( C \) = the set of students who play Cricket
- \( H \) = the set of students who play Hockey

### Step 2: Identify the given values
From the problem, we know:
- Total number of students = 45
- Number of students who play Cricket (\( |C| \)) = 30
- Number of students who play Hockey (\( |H| \)) = 27

### Step 3: Use the formula for the union of two sets
We know that:
\[ |C \cup H| = |C| + |H| - |C \cap H| \]
Where \( |C \cup H| \) is the total number of students playing either Cricket or Hockey, and \( |C \cap H| \) is the number of students who play both sports.

### Step 4: Substitute the known values into the formula
Substituting the values we have:
\[ 45 = 30 + 27 - |C \cap H| \]

### Step 5: Solve for \( |C \cap H| \)
Now, simplify the equation:
\[ 45 = 57 - |C \cap H| \]
Rearranging gives:
\[ |C \cap H| = 57 - 45 \]
\[ |C \cap H| = 12 \]

### Step 6: Calculate the number of students who play Cricket only
The number of students who play Cricket only can be calculated using:
\[ \text{Students who play Cricket only} = |C| - |C \cap H| \]
Substituting the known values:
\[ \text{Students who play Cricket only} = 30 - 12 = 18 \]

### Final Answer
Thus, the number of students who play only Cricket is **18**.

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