In a marraiage party total 278 guests were present 20 guests took Pepsi and Dew, 23 guests took Dew and Sprite and 21 guests took Pepsi, and Sprite and 9 guesta took all the three cold drinks viz. Pepsi, Sprite and Dew. It is also known that there were equal number of bottles of each of three kinds viz. Pepsi, Dew and Sprite .
(a) How many guests took only one bottle colddrink ?
(b) How many guests enjoyed Dew or Sprite but not Pespsi ?
(c) How many guests enjoyed Pepsi and Dew but not Sprite .

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and Venn diagrams to organize the information.

### Step 1: Define the Variables
Let:
- \( P \): Number of guests who took Pepsi
- \( D \): Number of guests who took Dew
- \( S \): Number of guests who took Sprite
- \( x \): Number of guests who took only Pepsi
- \( y \): Number of guests who took only Dew
- \( z \): Number of guests who took only Sprite
- \( a \): Number of guests who took only Pepsi and Dew
- \( b \): Number of guests who took only Dew and Sprite
- \( c \): Number of guests who took only Pepsi and Sprite
- \( d \): Number of guests who took all three drinks (Pepsi, Dew, Sprite)

From the problem, we have:
- \( d = 9 \) (guests who took all three drinks)
- \( a + d = 20 \) (guests who took Pepsi and Dew)
- \( b + d = 23 \) (guests who took Dew and Sprite)
- \( c + d = 21 \) (guests who took Pepsi and Sprite)

### Step 2: Calculate the Values of a, b, and c
Using the equations from above:
1. \( a = 20 - d = 20 - 9 = 11 \)
2. \( b = 23 - d = 23 - 9 = 14 \)
3. \( c = 21 - d = 21 - 9 = 12 \)

### Step 3: Set Up the Total Equation
The total number of guests can be expressed as:
\[ x + y + z + a + b + c + d = 278 \]
Substituting the known values:
\[ x + y + z + 11 + 14 + 12 + 9 = 278 \]
This simplifies to:
\[ x + y + z + 46 = 278 \]
Thus,
\[ x + y + z = 278 - 46 = 232 \]

### Step 4: Calculate the Number of Guests Who Took Only One Drink
We need to find \( x + y + z \):
- \( x \): Guests who took only Pepsi
- \( y \): Guests who took only Dew
- \( z \): Guests who took only Sprite

From the previous step, we already calculated:
\[ x + y + z = 232 \]

### Step 5: Find the Number of Guests Who Enjoyed Dew or Sprite but Not Pepsi
This includes guests who took:
- Only Dew: \( y \)
- Only Sprite: \( z \)
- Both Dew and Sprite but not Pepsi: \( b = 14 \)

Thus, the total is:
\[ y + z + b = y + z + 14 \]
We already know \( y + z = 232 - x \). To find \( x \), we can use the total number of guests:
\[ x + (y + z + 14) = 278 - d \]
Substituting \( d \):
\[ x + (232 - x + 14) = 278 - 9 \]
This simplifies to:
\[ 232 + 14 = 269 \]
Thus, we find:
\[ y + z + 14 = 232 - x + 14 \]

### Step 6: Calculate Guests Who Enjoyed Pepsi and Dew but Not Sprite
This is simply \( a \):
\[ a = 11 \]

### Final Answers
(a) The number of guests who took only one cold drink: **232**
(b) The number of guests who enjoyed Dew or Sprite but not Pepsi: **y + z + 14 = 232 - x + 14**
(c) The number of guests who enjoyed Pepsi and Dew but not Sprite: **11**