A class has 30 students. The following prizes are to be awarded to the students of this class. First and second in Mathematics; first and second in physics first in chemistry and first in biology. If N denote the number of ways in which this can be done, then

First and second prizes in Maths and Physics can be awarded in

${}^{30}{P}_2{.}^{30}{P}_2={\left({}^{30}{P}_2\right)}^2$  ways

First prize is Chemistry, Biology can be awarded in $=30.30={\left(30\right)}^2$  ways

$\therefore$  $N={\left({}^{30}{P}_2\right)}^2.{\left(30\right)}^2={30}^4{.29}^2=2^4{.3}^4{.5}^4{.29}^2$

As $400=2^4{.5}^2,600=2^3{\mathrm{.3.5}}^2,8100=2^2{.3}^4{.5}^2,$

We get N is divisible by each of $400,600,8100$

and also N is divisible by 4 primes 2, 3, 5, 29