TITLE: Why doesn't the multiplication rule work here?
QUESTION [5 upvotes]: I have the following question:

How many ways are there to select $3$ candidates from $8$ equally
qualified recent graduates for openings in an accounting firm?

I was given the following solution to it: $\binom{8}{3}$, however, I got the answer: $8 \times 7 \times 6$ by doing the multiplication rule, I was wondering why my answer doesn't work.
My initial reason is that in the multiplication rule there is the underlying assumption that order matters and here obviously order doesn't matter. However, I am posting this here as that is the first time I come to such a realization and I am not sure that my reasoning is correct, especially since that I never learned multiplication rule with the idea of order in my mind. Any clarification would be appreciated.

REPLY [2 votes]: I think your question seems to be when you can assume order doesn't matter, and when it actually does.
I'll give you two examples which might make it easier.
The first is like your own question. How many ways of choosing 3 people out of 8 people? You're just choosing people, and not assigning any special role or task to them to distinguish between individual members of your chosen triple. Here, use $^8C_3$.
The second is a variation of your question. How many ways of choosing 3 club office holders (namely President, Secretary and Treasurer) out of 8 people? From the wording, you're not just choosing three people, you're also assigning each to a particular role. In this case, order definitely matters since $(a, b, c)$ in the respective roles of President, Secretary and Treasurer is a very different selection from $(c, b, a)$ in those respective roles. So here you use $^8P_3$ (which is equal to $8\cdot 7 \cdot 6$.