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

# 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

1. A
$400|N$
2. B
$600|N$
3. C
$8100|N$
4. D
$N$ is divisible by four distinct prime numbers

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### Solution:

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 $400={2}^{4}{.5}^{2},\text{\hspace{0.17em}}600={2}^{3}{.3.5}^{2},\text{\hspace{0.17em}}8100={2}^{2}{.3}^{4}{.5}^{2},$

We get N is divisible by each of $400,\text{\hspace{0.17em}}600,\text{\hspace{0.17em}}8100$

and also N is divisible by 4 primes 2, 3, 5, 29  +91

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