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

• A. N is divisible by 400
• B. N is divisible by 600
• C. N is divisible by 8100
• D. N is divisible by four distinct prime numbers

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

N is divisible by 400

B

N is divisible by 600

C

N is divisible by 8100

D

N is divisible by four distinct prime numbers

First and second prizes in Mathematics (Physics) can be awarded in ${}^{30}{P}_2{(}^{30}{P}_2)$ ways. First prize in Chemistry (Biology) can be awarded in 30 (30) ways. Therefore, $N={(}^{30}{P}_2{)}^2\left({30}^2\right)={30}^4{29}^2=2^4.3^4.5^4.{29}^2$. Since 400 $=2^4{.5}^2,600=2^3.3{.5}^2$ and 8100 $=2^2{.3}^4.5^2$ we get N is divisible by each of 400, 600 and 8100. Also N is divisible by four distinct primes. viz. 2, 3, 5 and 29.