Not able to understand the given problem.

Question

In a certain college at the B.Sc. examination, $3$ candidates obtained first class honours in each of the following subjects: Physics, Chemistry and Maths, no candidates obtaining honours in more than one subject; 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 in any one subject is

I am having difficulty in understanding this problem.

What I understood :

There are $3$ candidates, one of them obtained obtained first class honour in Maths, another obtained first class honour in Chemistry and another one obtained first color honour in Physics

Now I am not getting the connection of these $3$ candidates with $9$ candidates mentioned. Please help me in this.

Answer 1

For each of the three subjects, three candidates obtained honors in that subject. This makes nine total candidates.

Answer 2

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 in any one subject is

There are 9 candidates and 9 scholarships and 3 subjects. For each subject, there are 3 candidates who have obtained different positions/marks. Now, we have to distribute 9 scholarships, which have different values, to the 9 candidates.

From here, we can see that the answer is
$$\frac{(9-1)!}{4!}=1680$$

Answer 3

I believe three candidates achieved first class honours in physics, three in chemistry, and three in maths. It appears that within each subject, the students are ranked. So one of the physics candidates did best, one did second best, and the remaining candidate performed worst. The rule is that a better performing student must receive a more valuable scholarship.

The key to this problem is that once you have decided which subjects are to receive which scholarships, how to assign the scholarships to the candidates in the subject is completely fixed. The makes a multinomial coefficient problem, or equivalently, a problem of letter arrangements with repeated letters. (The letters will represent the subjects, and the order of the letters will represent the assignment of subjects to scholarships.)