Out of 35 students in section A of the `7^(th)` grade of Manhattan Public School , 10 students like baseball, 20 students like football and 10 students like rugby. 3 students like baseball and football,2 students like only basketball and rugby,4 students like only baseball and rugby. If only 2 students like all three games , how many students do not like any of the above three games ?

Options

A. 5
B. 6
C. 8
D. 9

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and a Venn diagram approach.

### Step 1: Define the Sets
Let:
- \( A \) = Set of students who like baseball
- \( B \) = Set of students who like football
- \( C \) = Set of students who like rugby

From the problem, we have:
- \( |A| = 10 \) (students like baseball)
- \( |B| = 20 \) (students like football)
- \( |C| = 10 \) (students like rugby)

### Step 2: Identify the Overlaps
We are given the following overlaps:
- \( |A \cap B| = 3 \) (students like both baseball and football)
- \( |B \cap C| = 2 \) (students like both football and rugby)
- \( |A \cap C| = 4 \) (students like both baseball and rugby)
- \( |A \cap B \cap C| = 2 \) (students like all three sports)

### Step 3: Calculate the Exclusive Counts
Now we need to calculate the number of students who like only one sport:

1. **Only Baseball**:
\[ |A \text{ only}| = |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 10 - (3 + 4 - 2) = 10 - 5 = 5 \]

2. **Only Football**:
\[ |B \text{ only}| = |B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 20 - (3 + 2 - 2) = 20 - 3 = 17 \]

3. **Only Rugby**:
\[ |C \text{ only}| = |C| - (|A \cap C| + |B \cap C| - |A \cap B \cap C|) = 10 - (4 + 2 - 2) = 10 - 4 = 6 \]

### Step 4: Calculate Total Students Who Like at Least One Sport
Now we can calculate the total number of students who like at least one sport:
\[ \text{Total who like at least one sport} = |A \text{ only}| + |B \text{ only}| + |C \text{ only}| + |A \cap B| + |A \cap C| + |B \cap C| + |A \cap B \cap C| \]
Substituting the values:
\[ = 5 + 17 + 6 + 3 + 4 + 2 + 2 = 39 \]

### Step 5: Calculate Students Who Do Not Like Any Sport
The total number of students is 35. Therefore, the number of students who do not like any of the sports is:
\[ \text{Students who do not like any sport} = \text{Total students} - \text{Total who like at least one sport} \]
\[ = 35 - 27 = 8 \]

### Final Answer
Thus, the number of students who do not like any of the above three games is **8**.

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