The number of prime numbers among the numbers `105! + 2, 105! +3, 105! +4,...., 105! + 104, 105!+105,` is
(a) 31 (b) 32 (c) 33 (d) None of these

Options

A. 31
B. 32
C. 33
D. None of these

Answer

To determine the number of prime numbers among the numbers \( 105! + 2, 105! + 3, \ldots, 105! + 105 \), we can analyze each term in the sequence.

### Step-by-Step Solution:

1. **Understanding Factorials**:
The expression \( 105! \) (105 factorial) is the product of all positive integers from 1 to 105. Therefore, \( 105! \) is divisible by every integer from 1 to 105.

2. **Analyzing Each Term**:
We will analyze each term \( 105! + k \) where \( k \) ranges from 2 to 105.

3. **Checking \( 105! + 2 \)**:
- \( 105! \) is divisible by 2 (since it includes the factor 2).
- Therefore, \( 105! + 2 \) is also even and greater than 2, which means it cannot be prime.

4. **Checking \( 105! + 3 \)**:
- \( 105! \) is divisible by 3 (since it includes the factor 3).
- Thus, \( 105! + 3 \) is also divisible by 3 and greater than 3, so it cannot be prime.

5. **Continuing the Pattern**:
- This reasoning can be applied to all integers \( k \) from 2 to 105.
- For any \( k \) in this range, \( 105! + k \) will be divisible by \( k \) since \( 105! \) contains \( k \) as a factor.

6. **Final Analysis**:
- Since every term \( 105! + k \) (for \( k = 2, 3, \ldots, 105 \)) is divisible by \( k \) and greater than \( k \), none of these terms can be prime.

7. **Conclusion**:
- Therefore, the number of prime numbers among the numbers \( 105! + 2, 105! + 3, \ldots, 105! + 105 \) is **0**.

### Answer:
The correct option is (d) None of these.