In a group of 200 students, 20 played cricket only, 36 plated tennis only, 40 played, hockey only, 8 played cricket and tennis, 20 played cricket and hockey, 28 played hockey and tennis and 80 played hockey. By Venn diagram, find the number of students who (i) played cricket (ii) played tennis (iii) did not played any of the above three games.

Explanation

To solve the problem step by step, we will use a Venn diagram to represent the students playing cricket, tennis, and hockey.

### Step 1: Define the Variables
Let:
- \( C \): Set of students who played cricket
- \( T \): Set of students who played tennis
- \( H \): Set of students who played hockey

From the problem, we have the following information:
- \( n(C \cap T^c \cap H^c) = 20 \) (students who played cricket only)
- \( n(T \cap C^c \cap H^c) = 36 \) (students who played tennis only)
- \( n(H \cap C^c \cap T^c) = 40 \) (students who played hockey only)
- \( n(C \cap T) = 8 \) (students who played both cricket and tennis)
- \( n(C \cap H) = 20 \) (students who played both cricket and hockey)
- \( n(H \cap T) = 28 \) (students who played both hockey and tennis)
- \( n(H) = 80 \) (total students who played hockey)

### Step 2: Set Up the Venn Diagram
We will denote:
- \( x \): Students who played all three games (Cricket, Tennis, and Hockey).
- The intersections can be expressed as follows:
- Students who played Cricket and Tennis only: \( 8 - x \)
- Students who played Cricket and Hockey only: \( 20 - x \)
- Students who played Hockey and Tennis only: \( 28 - x \)

### Step 3: Calculate Total Students Who Played Hockey
The total number of students who played hockey can be expressed as:
\[ n(H) = n(H \cap C^c \cap T^c) + n(H \cap C \cap T^c) + n(H \cap C^c \cap T) + n(H \cap C \cap T) \]
Substituting the values we have:
\[ 80 = 40 + (20 - x) + (28 - x) + x \]
Simplifying this:
\[ 80 = 40 + 20 + 28 - x \]
\[ 80 = 88 - x \]
\[ x = 88 - 80 = 8 \]

### Step 4: Substitute \( x \) Back to Find Other Values
Now substituting \( x = 8 \) into the intersections:
- Students who played Cricket and Tennis only: \( 8 - 8 = 0 \)
- Students who played Cricket and Hockey only: \( 20 - 8 = 12 \)
- Students who played Hockey and Tennis only: \( 28 - 8 = 20 \)

### Step 5: Calculate Total Students Who Played Cricket and Tennis
Now we can calculate the total number of students who played cricket:
\[ n(C) = n(C \cap T^c \cap H^c) + n(C \cap T) + n(C \cap H) = 20 + 0 + 12 + 8 = 40 \]
And for tennis:
\[ n(T) = n(T \cap C^c \cap H^c) + n(T \cap C) + n(T \cap H) = 36 + 0 + 20 + 8 = 64 \]

### Step 6: Calculate Students Who Did Not Play Any Game
The total number of students is 200. The total number of students who played at least one game is:
\[ n(C) + n(T) + n(H) - (n(C \cap T) + n(C \cap H) + n(H \cap T) + n(C \cap T \cap H)) \]
Substituting the values:
\[ = 40 + 64 + 80 - (0 + 12 + 20 + 8) = 184 - 40 = 144 \]
Thus, the number of students who did not play any game:
\[ 200 - 144 = 56 \]

### Final Answers
1. Number of students who played Cricket: **40**
2. Number of students who played Tennis: **64**
3. Number of students who did not play any of the above games: **56**