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

$^{30} P_{2} .^{30} P_{2} = {(^{30} P_{2})}^{2}$ ways

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

$∴$ $N = {(^{30} P_{2})}^{2} . {(30)}^{2} = 30^{4} {.29}^{2} = 2^{4} {.3}^{4} {.5}^{4} {.29}^{2}$

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

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

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

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