division - combinations

Question

In how many ways can 20 recruits be distributed into 4 groups each consisting of 5 recruits? In how many ways can they be distributed into 4 camps, each camp receiving 5 recruits?

I have a conceptual doubt in this question .

So should it be $\displaystyle\frac{20!}{(5!)^4}$ for division into both groups and camps?

Why is it $\displaystyle\frac{20!}{4!(5!)^4}$ for division into groups? I can't understand what is the difference between camps and groups.

Is there a separate concept for such questions?

It looks as if the question assumes that the groups are indistinguishable(groups aren't labeled), while the camps are distinguishable(camps are labeled). — Aiden Chow, Aug 27 '20 at 05:15
The question is not stated well. You couldn't have known that camps and groups are different until you looked at the answer. — Aiden Chow, Aug 27 '20 at 05:26

score 1 · Answer 1 · answered Aug 27 '20 at 05:13

1

Presumably each camp has a name, while you don't care about which group is which. When you divide into camps, you pick five for the first camp, five for the second, and so on, which gives the expression you have. Then there are $4!$ of these divisions that give the same groups because the groups could have been chosen in any order.

answered Aug 27 '20 at 05:13

Ross Millikan

374,822

So I should assume camps are labelled even though, in this question they haven't mentioned that camps are labelled , right? – Dia Swamy Aug 27 '20 at 05:17
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Yes, I would read the question that way. By having both camps and groups there has to be something different about them or you wouldn't ask both parts. Because camps exist before you assign the recruits they are probably different in some way. It would be clearer to spell that out. – Ross Millikan Aug 27 '20 at 05:20
oh, so camps exist before assigning recruits, alright . Got it. – Dia Swamy Aug 27 '20 at 05:21

division - combinations

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