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?

1.) First and second in Mathematics:

There are 30 students eligible for the first prize and 29 students remaining for the second prize.

So, total number of ways of distributing the prize = $$30\times\ 29$$

2.) First and second in Physics:

There are 30 students eligible for the first prize and 29 students remaining for the second prize.

So, total number of ways of distributing the prize = $$30\times\ 29$$

3.) First in Chemistry:

There are 30 students eligible for the first prize.

So, total number of ways of distributing the prize = 30

4.) First in English:

There are 30 students eligible for the first prize.

So, total number of ways of distributing the prize = 30

So, total number of ways of giving all the prizes = $$\left(30\times\ 29\right)\times\ \left(30\times\ 29\right)\times\ 30\times\ 30=30^4\times\ 29^2$$.

So, correct answer is option D.