Can someone explain the logic of combinations in this question? Problem:

An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs, each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to choose the leadership of the tribe? That is, in how many ways can we choose a chief, 2 supporting chiefs, and two inferior officers reporting to each supporting chief?

Source: AoPS Alcumus
Given Solution:

There are 10 choices for the chief. For each choice, there are $\binom{9}{2}$ to choose the 2 supporting chiefs from the remaining members, for a total of $10\cdot \binom{9}{2}$ ways to choose the chief and supporting chiefs. There are then $\binom{7}{2}$ ways to choose the inferior officers for the first supporting chief and $\binom{5}{2}$ ways to choose the inferior officers for the second supporting chief. This gives us a total of $10\cdot\binom{9}{2}\cdot \binom{7}{2}\cdot\binom{5}{2} = \boxed{75600}$ ways to form the leadership of the tribe.

My questions:


*

*How can they do $\binom{9}{2}$ for the first level chiefs when it is not given that they are indistinct? I would normally go about $^9P_2=9\cdot8$ i.e. 9 choices for the first and 8 choices for the second chief.

*Why it that we do $\binom{7}{2}\cdot\binom{5}{2}$ for the inferior officers? Why not $\binom{7}{4}$?


I am sorry but I am new to this stuff so these might seem obvious to you but are not so for me. Thus, can some one please explain the answers for my questions exactly? Thanks!
 A: "How can they do ${9 \choose 2}$ when it's not given they are indistinct?"
I think they are indicating that they are indistinct when they distinguish both as "supporting chiefs".  It's implied that if Bob is the head chief and Martha and Taylor are the support chiefs is the same thing as Bob being the head chief and Taylor and Martha are the support chiefs.
"Why ${7 \choose 5} \cdot {5 \choose 3}$ and not ${7 \choose 4}$?"
Because two inferior officers work for specific support chiefs and the pairs are distinguishable.  If Sheila and Conception work for Martha while Enrico and Clyde work for Taylor, this is different than if Sheila and Enrico work for Martha while Conception and Clyde work for Taylor, even though those are the same 4 inferior officers.
In short for m out of n:  
If order matters:  n*(n-1)....(n-m+1) = n!/(n-m)!
If order doesn't matter:  [n!/(n-m)!]/m! = ${n \choose m}$
If there are two groups r and v and order doesn't matter: ${n \choose r}*{{n -r} \choose v}$.
And If there are two groups r and v and order does matter: $n!/(n - r)!*(n - r)!/(n - r -v)! = n!/m!$. (Note: if order matters size and number of groups do not!)
A: On point (1), if the original problem had specified "2 equal supporting chiefs"
instead of just "2 supporting chiefs," it would be easier to see that the intended solution was to regard the two supporting chiefs as interchangeable,
that is, "Chief Mary with supporting chiefs Tom and Ed" would be the same as
"Chief Mary with supporting chiefs Ed and Tom."
But consider the wording at the very end of the original question: 
"in how many ways can we choose a chief, 2 supporting chiefs, and two inferior officers reporting to each supporting chief?"
If we just insert the words "from a tribe of ten members" in an
appropriate place, the end of the question is a complete question in itself,
and we can regard everything that came before it as material to help
understand what we mean by those words. That is, when the problem identifies
two officers in a particular position, their roles are interchangeable.
Obviously, the wording of the whole problem is not completely consistent
and logical. The trick is to extract the most obvious meaning.
As for part (2), I see fleablood already answered that part while I was
working on this.
A: For question 1, I would venture to say that people (chiefs in this case) are by nature distinct. For question 2, you could pick four inferior officers all at once, but then you would have to choose two of them for inferior chief A and two for inferior chief B.
