In a group of 132 people 50, 60, 70 people like three different sweets-Barfi, Jalebi, Rasgulla, respectively. The number of people who like all the three sweets is half the number of people who like exactly two sweets. The number of people who like exactly any two out of the three sweets is the same as those who like exactly any other two of the three sweets. The number of people who like the three sweets

Options

A. 12
B. 6
C. 8
D. none of these

Answer

To solve the problem step by step, we will use the information given and set up equations based on the relationships described.

### Step 1: Define Variables
Let:
- \( F \) = Number of people who like all three sweets (Barfi, Jalebi, Rasgulla)
- \( D \) = Number of people who like exactly Barfi and Jalebi
- \( E \) = Number of people who like exactly Barfi and Rasgulla
- \( G \) = Number of people who like exactly Jalebi and Rasgulla

### Step 2: Set Up Equations
From the problem, we know:
1. The number of people who like all three sweets is half the number of people who like exactly two sweets:
\[ F = \frac{1}{2}(D + E + G) \]

2. The number of people who like exactly any two out of the three sweets is the same:
\[ D = E = G \]
Let \( D = E = G = d \). Thus, we can rewrite the first equation:
\[ F = \frac{1}{2}(d + d + d) = \frac{3d}{2} \]

### Step 3: Total People Who Like Each Sweet
We are given:
- People who like Barfi = 50
- People who like Jalebi = 60
- People who like Rasgulla = 70

Using the principle of inclusion-exclusion, we can express the total number of people who like each sweet:
1. For Barfi:
\[ 50 = B + D + E + F \]
Substituting \( E \) and \( D \) with \( d \):
\[ 50 = B + d + d + F \implies 50 = B + 2d + F \]

2. For Jalebi:
\[ 60 = J + D + G + F \implies 60 = J + d + d + F \implies 60 = J + 2d + F \]

3. For Rasgulla:
\[ 70 = R + E + G + F \implies 70 = R + d + d + F \implies 70 = R + 2d + F \]

### Step 4: Summing the Equations
Now, we can add the three equations:
\[ (50 + 60 + 70) = (B + J + R) + 3d + 3F \]
This simplifies to:
\[ 180 = (B + J + R) + 3d + 3F \]

### Step 5: Total Number of People
We also know the total number of people is 132:
\[ 132 = B + J + R + D + E + F + G = B + J + R + 3d + F \]

### Step 6: Solve the Equations
Now we have two equations:
1. \( 180 = (B + J + R) + 3d + 3F \)
2. \( 132 = (B + J + R) + 3d + F \)

Subtract the second equation from the first:
\[ 180 - 132 = 3F - F \implies 48 = 2F \implies F = 24 \]

### Step 7: Find \( d \)
Now substitute \( F \) back into the equation for \( F \):
\[ F = \frac{3d}{2} \implies 24 = \frac{3d}{2} \implies 3d = 48 \implies d = 16 \]

### Conclusion
The number of people who like all three sweets (Barfi, Jalebi, Rasgulla) is:
\[ \boxed{24} \]