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 Cricket only?
A. 11
B. 13
C. 8
D. 5

Options

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

Answer

To find out how many students play Cricket only, we can use the principle of inclusion-exclusion and Venn diagrams. Let's break down the problem step by step.

### Step 1: Define the sets
Let:
- \( C \) = number of students who play Cricket = 25
- \( F \) = number of students who play Football = 30
- \( V \) = number of students who play Volleyball = 24

### Step 2: Define the intersections
We also know:
- \( C \cap F \) = number of students who play both Cricket and Football = 10
- \( C \cap V \) = number of students who play both Cricket and Volleyball = 9
- \( F \cap V \) = number of students who play both Football and Volleyball = 12
- \( C \cap F \cap V \) = number of students who play all three sports = 5

### Step 3: Calculate the number of students playing only two sports
Now, we can find the number of students who play only two sports:
- Students who play Football and Volleyball but not Cricket:
\[ (F \cap V) - (C \cap F \cap V) = 12 - 5 = 7 \]
- Students who play Cricket and Football but not Volleyball:
\[ (C \cap F) - (C \cap F \cap V) = 10 - 5 = 5 \]
- Students who play Cricket and Volleyball but not Football:
\[ (C \cap V) - (C \cap F \cap V) = 9 - 5 = 4 \]

### Step 4: Calculate the total number of students playing Cricket
Now, we can calculate the total number of students who play Cricket:
- Students who play only Cricket:
Let \( x \) be the number of students who play only Cricket.
The total number of students who play Cricket can be expressed as:
\[ x + (C \cap F \cap V) + (C \cap F) - (C \cap F \cap V) + (C \cap V) - (C \cap F \cap V) = C \]
Substituting the known values:
\[ x + 5 + 5 + 4 = 25 \]
Simplifying gives:
\[ x + 14 = 25 \]
Therefore:
\[ x = 25 - 14 = 11 \]

### Conclusion
The number of students who play Cricket only is **11**.