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  • 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 i

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

 

Option: 1

400 \mid N


Option: 2

600 \mid N


Option: 3

8100 \mid N

 


Option: 4

N is divisible by four distinct prime numbers


Answers (1)

best_answer

First and second prizes in Mathematics (Physics) can be awarded in \left({ }^{30} P_2\right)\left({ }^{30} P_2\right)=\left({ }^{30} P_2\right)^2=(30)^2(29)^2 ways. First prize in Chemistry (Biology) can be awarded in \left({ }^{30} P_1\right)\left({ }^{30} P_1\right)=(30)^2 ways.

Hence, 

\begin{array}{r} N=(30)^2(29)^2(30)^2=(30)^4(29)^2=2^4 \cdot 3^4 \cdot 5^4 \cdot 29^2 \\ \\\text { Since, } \quad 400=2^4 \cdot 5^2, 600=2^3 \cdot 3 \cdot 5^2, 8100=2^2 \cdot 3^4 \cdot 5^2 \end{array}

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

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

Posted by

Sanket Gandhi

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