In a group of 24 children , each one play cricket or hockey or both. If 16 play cricket and 12 play cricket only , find how many play hockey only.

Answer

To solve the problem step by step, we will use the information provided about the number of children playing cricket and hockey.

### Step 1: Define the Sets
Let:
- \( A \) be the set of children who play cricket.
- \( B \) be the set of children who play hockey.

### Step 2: Identify Given Information
From the problem, we know:
- Total number of children, \( N(A \cup B) = 24 \)
- Number of children who play cricket, \( N(A) = 16 \)
- Number of children who play cricket only, \( N(A - B) = 12 \)

### Step 3: Find the Number of Children Who Play Both Sports
The number of children who play both cricket and hockey can be calculated using the formula:
\[ N(A) = N(A - B) + N(A \cap B) \]
Where:
- \( N(A \cap B) \) is the number of children who play both sports.

Substituting the known values:
\[ 16 = 12 + N(A \cap B) \]
To find \( N(A \cap B) \):
\[ N(A \cap B) = 16 - 12 = 4 \]

### Step 4: Use the Union Formula
Now we can use the formula for the union of two sets:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]
We know \( N(A \cup B) = 24 \), \( N(A) = 16 \), and \( N(A \cap B) = 4 \). We need to find \( N(B) \):
\[ 24 = 16 + N(B) - 4 \]
Simplifying this gives:
\[ 24 = 12 + N(B) \]
So,
\[ N(B) = 24 - 12 = 12 \]

### Step 5: Find the Number of Children Who Play Hockey Only
Now, we can find the number of children who play hockey only using:
\[ N(B - A) = N(B) - N(A \cap B) \]
Substituting the known values:
\[ N(B - A) = 12 - 4 = 8 \]

### Final Answer
Thus, the number of children who play hockey only is **8**.

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