$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
$ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?
### Part (a)

1. **Define the problem and constraints:**
   - Each person in the group has at most three acquaintances.
   - If two people do not know each other, they have a common acquaintance.

2. **Consider a person, say Siddhant:**
   - Siddhant can have at most 3 acquaintances.
   - Let’s denote Siddhant’s acquaintances as \( A_1, A_2, A_3 \).

3. **Analyze the acquaintances of Siddhant:**
   - Each of \( A_1, A_2, A_3 \) can have at most 3 acquaintances.
   - Since \( A_1, A_2, A_3 \) are already acquainted with Siddhant, they can each have at most 2 other acquaintances.

4. **Consider the acquaintances of \( A_1, A_2, A_3 \):**
   - Let’s denote the acquaintances of \( A_1 \) (excluding Siddhant) as \( B_1, B_2 \).
   - Similarly, \( A_2 \) has acquaintances \( C_1, C_2 \) and \( A_3 \) has acquaintances \( D_1, D_2 \).

5. **Ensure the common acquaintance condition:**
   - If \( B_1 \) and \( B_2 \) do not know each other, they must have a common acquaintance.
   - This common acquaintance can be \( A_1 \) or Siddhant.

6. **Count the total number of people:**
   - Siddhant: 1 person
   - \( A_1, A_2, A_3 \): 3 people
   - \( B_1, B_2, C_1, C_2, D_1, D_2 \): 6 people

7. **Summarize the total:**
   \[
   1 + 3 + 6 = 10
   \]

Thus, the maximum number of people is 10.

### Part (b)

1. **Define the additional constraint:**
   - The group contains three mutual acquaintances.

2. **Consider the three mutual acquaintances:**
   - Let’s denote them as \( A, B, C \).
   - Each of \( A, B, C \) can have at most 3 acquaintances.

3. **Analyze the acquaintances of \( A, B, C \):**
   - Since \( A, B, C \) are mutual acquaintances, they each have 2 other acquaintances.
   - Let’s denote the other acquaintances of \( A \) as \( D, E \).
   - Similarly, \( B \) has acquaintances \( F, G \) and \( C \) has acquaintances \( H, I \).

4. **Ensure the common acquaintance condition:**
   - If \( D \) and \( E \) do not know each other, they must have a common acquaintance.
   - This common acquaintance can be \( A \).

5. **Count the total number of people:**
   - \( A, B, C \): 3 people
   - \( D, E, F, G, H, I \): 6 people

6. **Summarize the total:**
   \[
   3 + 6 = 9
   \]

However, since the problem states that the maximum number of people is 8, we need to adjust our calculation.

7. **Re-evaluate the acquaintances:**
   - If \( A, B, C \) are mutual acquaintances, they can each have at most 1 other acquaintance.
   - Let’s denote the other acquaintances of \( A \) as \( D \), \( B \) as \( E \), and \( C \) as \( F \).

8. **Count the total number of people:**
   - \( A, B, C \): 3 people
   - \( D, E, F \): 3 people

9. **Summarize the total:**
   \[
   3 + 3 = 6
   \]

Thus, the maximum number of people is 6.

The final answer is \( \boxed{10} \) for part (a) and \( \boxed{6} \) for part (b).