In how many ways can the following prizes be given away to a class of 30 students, first and second in Mathematics, first and second in Physics, first in Chemistry and first in English?

Options

A. `(30!)/(4!)`
B. `(30 )^4 xx(29)^2`
C. `(30)^3-1`
D. `(30)^4 xx(29 )^4`

Answer

To solve the problem of how many ways the prizes can be given away to a class of 30 students, we can break it down step by step.

### Step-by-Step Solution:

1. **Identify the Prizes and Students:**
- We have a total of 30 students.
- The prizes to be awarded are:
- First and second in Mathematics
- First and second in Physics
- First in Chemistry
- First in English

2. **Calculate the Ways to Award Mathematics Prizes:**
- For the first prize in Mathematics, any of the 30 students can win. So, there are 30 choices.
- After the first prize is awarded, only 29 students are left for the second prize. So, there are 29 choices for the second prize.
- Therefore, the total ways to award Mathematics prizes = \(30 \times 29\).

3. **Calculate the Ways to Award Physics Prizes:**
- Similarly, for the first prize in Physics, any of the 30 students can win, giving us 30 choices.
- After awarding the first prize, 29 students remain for the second prize. So, there are 29 choices for the second prize.
- Therefore, the total ways to award Physics prizes = \(30 \times 29\).

4. **Calculate the Ways to Award Chemistry Prize:**
- For the Chemistry prize, only the first prize is awarded. Any of the 30 students can win this prize, so there are 30 choices.
- Therefore, the total ways to award Chemistry prizes = \(30\).

5. **Calculate the Ways to Award English Prize:**
- For the English prize, only the first prize is awarded. Any of the 30 students can win this prize, so there are 30 choices.
- Therefore, the total ways to award English prizes = \(30\).

6. **Combine All the Choices:**
- To find the total number of ways to award all the prizes, we multiply the number of choices for each subject:
\[ \text{Total Ways} = (30 \times 29) \times (30 \times 29) \times 30 \times 30 \]

7. **Calculate the Final Result:**
- This simplifies to:
\[ \text{Total Ways} = (30 \times 29)^2 \times 30^2 \]
- Now, calculating:
\[ = 870 \times 870 \times 900 = 756300 \]

### Final Answer:
The total number of ways to award the prizes is **756300**.

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