Question

In a certain college at the B.Sc., examination in each subjects Physics, Chemistry, and Mathematics there are 3 students getting honors, no candidates obtaining honors in more than one subject; Find the number of ways in which 9 scholarships of different value be awarded to the 9 candidates if due regard is to be paid only to the places obtained by candidates on any one object.

Answer 1

In this question we have to find the number of ways of $9$ scholarships of different value with $9$ candidates. For that, we solve this by using a number of formulas. And also we are going to divide the total number of ways by multiplication of the number of ways of giving scholarship to three subjects. It is further solved in a complete step by step solution.

Formulas used:
The number of ways $n$ items can be arranged is given by the formula $n!$.

Complete step by step answer:
It is given that three candidates obtained first-class honors in Mathematics, physics, and chemistry.
Also given that no candidates obtain honors in more than one subject.
Therefore the possible number of ways giving scholarship in Mathematics $= 3!$
Similarly the number of ways of giving scholarship in physics $= 3!$
And the number of ways of giving scholarships in chemistry $= 3!$
It is also given that there are $9$ scholarships therefore the number of ways the scholarship can be awarded is $9!$
Here we have to find no. of. Ways in which $9$ scholarships of different values are awarded to $9$ candidates if due regard is to be paid only to the places obtained by candidates in any one subject.
The number of ways the scholarships can be offered is given by dividing the total number of ways scholarships awarded by the product of the respective number of ways each subject offered.
That is the required number of ways = $\dfrac{{9!}}{{3!3!3!}}$
$=1680$

The number of ways 9 scholarships can be awarded to 9 candidates is $1680$.

Note:
We have to divide the total number of possible ways by the product of the number of possible ways that scholarship can be awarded to each subject to get the final answer because of the fact that one candidate can get only one honor. Suppose if it is not given the result will vary.