In a statistical investigation of 1003 families of Calcutta, it was found that 63 families has neither a radio nor a T.V, 794 families has a radio and 187 has T.V. The number of families in that group having both a radio and a T.V is

Options

A. 36
B. 41
C. 32
D. None of these

Answer

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

### Step 1: Identify the total number of families and those with neither a radio nor a TV.
We know that:
- Total families surveyed = 1003
- Families with neither a radio nor a TV = 63

### Step 2: Calculate the number of families with either a radio or a TV.
To find the number of families that have either a radio or a TV, we can use the formula:
\[ n(R \cup T) = \text{Total families} - \text{Families with neither} \]
Substituting the values:
\[ n(R \cup T) = 1003 - 63 = 940 \]

### Step 3: Use the principle of inclusion-exclusion.
We know:
- Families with a radio (n(R)) = 794
- Families with a TV (n(T)) = 187

According to the principle of inclusion-exclusion:
\[ n(R \cup T) = n(R) + n(T) - n(R \cap T) \]
Where \( n(R \cap T) \) is the number of families with both a radio and a TV.

### Step 4: Rearranging the formula to find families with both a radio and a TV.
We can rearrange the equation to solve for \( n(R \cap T) \):
\[ n(R \cap T) = n(R) + n(T) - n(R \cup T) \]
Substituting the known values:
\[ n(R \cap T) = 794 + 187 - 940 \]

### Step 5: Calculate the number of families with both a radio and a TV.
Now performing the calculation:
\[ n(R \cap T) = 981 - 940 = 41 \]

### Conclusion:
The number of families that have both a radio and a TV is **41**.

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