Permutations

Question

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

Moderate

Solution

First and second prizes in Mathematics (Physics) can be awarded in $\left({}^{30}{\mathrm{P}}_{2}\right)\left({}^{30}{\mathrm{P}}_{2}\right)={\left({}^{30}{\mathrm{P}}_{2}\right)}^{2}=\left(30{)}^{2}\right(29{)}^{2}$ ways

First prize in Chemistry (Biology) can be awarded in $\left({}^{30}{\mathrm{P}}_{1}\right)\left({}^{30}{\mathrm{P}}_{1}\right)=(30{)}^{2}$ ways

Hence

$\mathrm{N}=\left(30{)}^{2}\right(29{)}^{2}(30{)}^{2}=(30{)}^{4}(29{)}^{2}={2}^{4}\cdot {3}^{4}\cdot {5}^{4}\cdot {29}^{2}$

since ,$400={2}^{4}\cdot {5}^{2},600={2}^{3}\cdot 3\cdot {5}^{2},8100={2}^{2}\cdot {3}^{4}\cdot {5}^{2}$

$\therefore $N is divisible by 400, 600, and 8100.

Also, N is divisible by four distinct primes numbers 2,3, 5, 29

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