A class has 30 students. The followign prizes are to be awarded to the students of the class- first and second in mathematics, first and second in Physics, first in Chemistry and first Biology. If N denote the number of ways in which this can be done, then

Options

A. N is divisible by 400
B. N is divisivble by 600
C. N is divisivble by 8100
D. N is divisible by 4 distinct prime No.

Answer

To solve the problem of awarding prizes to students in a class of 30, we need to calculate the total number of ways to distribute the prizes as follows:

1. **Identify the Prizes**:
- First Prize in Mathematics (1st Math)
- Second Prize in Mathematics (2nd Math)
- First Prize in Physics (1st Physics)
- Second Prize in Physics (2nd Physics)
- First Prize in Chemistry (1st Chemistry)
- First Prize in Biology (1st Biology)

2. **Calculate the Number of Ways to Award Each Prize**:
- For the **First Prize in Mathematics**, we have **30** choices (since there are 30 students).
- For the **Second Prize in Mathematics**, we have **29** choices (one student has already won the first prize).
- For the **First Prize in Physics**, we again have **30** choices.
- For the **Second Prize in Physics**, we have **29** choices.
- For the **First Prize in Chemistry**, we have **30** choices.
- For the **First Prize in Biology**, we have **30** choices.

3. **Calculate the Total Number of Ways (N)**:
The total number of ways to distribute the prizes can be calculated by multiplying the number of choices for each prize:
\[ N = (30 \times 29) \times (30 \times 29) \times (30) \times (30) \]
Simplifying this, we get:
\[ N = (30 \times 29)^2 \times (30^3) \]

4. **Calculate the Value of N**:
- First, calculate \(30 \times 29 = 870\).
- Then, calculate \(N = 870^2 \times 30^3\).

5. **Determine Divisibility**:
Now we need to check if \(N\) is divisible by the given options:
- **Option A: Divisible by 400**
- **Option B: Divisible by 600**
- **Option C: Divisible by 8100**
- **Option D: Divisible by 4 distinct prime numbers**

To check divisibility:
- **Divisibility by 400**: \(400 = 2^4 \times 5^2\)
- **Divisibility by 600**: \(600 = 2^3 \times 3 \times 5^2\)
- **Divisibility by 8100**: \(8100 = 2^2 \times 3^4 \times 5^2\)
- **Distinct Prime Numbers**: The prime factors of \(N\) need to be identified.

6. **Final Calculation**:
After performing the calculations for each option:
- We find that \(N\) is divisible by all four options.

### Summary of the Solution:
The total number of ways \(N\) to award the prizes is given by:
\[ N = (30 \times 29)^2 \times (30^3) \]
And it is divisible by 400, 600, 8100, and 4 distinct prime numbers.