The following questions are based on the information given below:
Out of a group of 60 students, 25 play Cricket, 30 play Football, 24 play Volleyball, 10 play Cricket and Football, 9 play Cricket and Volleyball, 12 play Volleyball and Football and 5 play all the three.
How many students play exactly two games?
A. 7
B. 13
C. 5
D. 16

Options

A. D
B. B
C. C
D. A

Answer

To solve the problem of how many students play exactly two games, we can use the principle of inclusion-exclusion. Let's break down the information given and calculate step by step.

### Step 1: Define the Variables
Let:
- \( C \) = number of students playing Cricket = 25
- \( F \) = number of students playing Football = 30
- \( V \) = number of students playing Volleyball = 24
- \( CF \) = number of students playing both Cricket and Football = 10
- \( CV \) = number of students playing both Cricket and Volleyball = 9
- \( FV \) = number of students playing both Football and Volleyball = 12
- \( CFV \) = number of students playing all three games = 5

### Step 2: Calculate Students Playing Exactly Two Games
To find the number of students playing exactly two games, we need to subtract those who play all three games from each pair of games.

1. **Cricket and Football (CF)**:
\[ \text{Students playing exactly Cricket and Football} = CF - CFV = 10 - 5 = 5 \]

2. **Cricket and Volleyball (CV)**:
\[ \text{Students playing exactly Cricket and Volleyball} = CV - CFV = 9 - 5 = 4 \]

3. **Football and Volleyball (FV)**:
\[ \text{Students playing exactly Football and Volleyball} = FV - CFV = 12 - 5 = 7 \]

### Step 3: Sum Up All Students Playing Exactly Two Games
Now, we can add the number of students playing exactly two games:
\[ \text{Total students playing exactly two games} = (CF - CFV) + (CV - CFV) + (FV - CFV) = 5 + 4 + 7 = 16 \]

### Final Answer
Thus, the total number of students who play exactly two games is **16**.