Question

In how many ways can the following prizes be given away to a class of 20 students:
First and second in mathematics; first and second in chemistry; first in physics and first in English?

Answer 1

Hint: We will start by first finding the ways in which we can give first prize in mathematics to a student in a class of 20 students. Then except this student we will give second prize in mathematics to the remaining one student out of 19 students. Then will similarly find another prize.

Complete step-by-step answer:
Now, we have been given a class of 20 students.
Now, first we have to give first and second prize in mathematics. So, the first prize can be given to any one of 20 students and second to any one of the remaining 19 students. Therefore, the ways for giving first and second prize in mathematics is $19\times 20$.
Now, similarly for first and second in chemistry we have the number of ways as $19\times 20$.
Now, similarly we have the ways in which we can give first in physics is 20.
Also, the ways of giving first in English is 20.
Therefore, the total ways in which we can give all prizes can be found by multiplication rule as $\begin{aligned} & \left( 20\times 19 \right)\times \left( 20\times 19 \right)\times 20\times 20 \\\ & ={{20}^{4}}\times {{19}^{2}} \\\ \end{aligned}$
Therefore, the total amount is $57,760,000$.

Note: It is important to note that once, we have given first prize in mathematics to a student then the same student cannot be given second prize in mathematics but can be given a prize in other subject also.