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 Mathematics (Physics) can be awarded in  30P2 30P2= 30P22=(30)2(29)2 waysFirst prize in Chemistry (Biology) can be awarded in  30P1 30P1=(30)2 ways Hence N=(30)2(29)2(30)2=(30)4(29)2=24⋅34⋅54⋅292since ,400=24⋅52,600=23⋅3⋅52,8100=22⋅34⋅52∴N is divisible by 400, 600, and 8100. Also, N is divisible by four distinct primes numbers 2,3, 5, 29