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 Physics, first in Chemistry and first in English?

Answer 1

Hint: To solve this question, we will start with giving first prize in Mathematics and second prize in Mathematics, to do that we will only take \[20 \times 19\], because only two students are getting prizes out of \[20\]. Similarly, we will give prizes in other subjects as well, to the \[20\] students of a class. Then, afterwards by multiplication rule, we will get the total number of ways of distributing prizes to the students.

Complete step-by-step answer:
We have been given that there is a class having \[20\] students. First and second prize should be given in Mathematics.
So, for that first prize be given to one student out of \[20.\] And second prize be given to students out of \[19.\]
Number of ways in giving first and second prize in Mathematics \[ = {\text{ }}20 \times 19\]
Similarly, Number of ways in giving first and second prize in Physics \[ = {\text{ }}20 \times 19\]
Now, Number of ways in giving first and prize in Chemistry \[ = {\text{ }}20\]
And, Number of ways in giving first and prize in English \[ = {\text{ }}20\]
So, by multiplication rule, the total number of ways the prize be given \[ = {\text{ }}(20 \times 19){\text{ }} \times {\text{ }}(20 \times 19){\text{ }} \times 20 \times 20\]
\[\begin{gathered}
   = {\text{ }}{20^4} \times {19^2} \\
   = {\text{ }}57760000 \\
\end{gathered} \]
Thus, in \[57760000\] ways the prizes can be given away to a class of \[20\] students.

Note: Here in this question, students should keep in mind while solving, that if one student received first prize in some subject, he can’t get second prize in that subject again because he already got first prize in that particular subject. But he can get prizes in other subjects.