First slide

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

A

400|N
B

600|N
C

8100|N
D

N is divisible by four distinct prime numbers

Solution

First and second prizes in Mathematics (Physics) can be awarded in $(^{30} P_{2}) (^{30} P_{2}) = {(^{30} P_{2})}^{2} = (30)^{2} (29)^{2}$ ways
First prize in Chemistry (Biology) can be awarded in $(^{30} P_{1}) (^{30} P_{1}) = (30)^{2}$ ways
Hence
$N = (30)^{2} (29)^{2} (30)^{2} = (30)^{4} (29)^{2} = 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 29^{2}$
since , $400 = 2^{4} \cdot 5^{2}, 600 = 2^{3} \cdot 3 \cdot 5^{2}, 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$
$∴$ 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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