Lets say I have 20k people.(k>0) Lets say 15k boys 5k girls. If I want to make k groups of 15 boys and 5 girls, where inner ordering of the group does matter(it can be girl boy boy ... or whatever) - point is the boys and girls are mixed within a group. How do I do it?
From my understanding 15 boys,then 5 girls in the following way 15k,15k-1,...15k-14,5k,5k-1,...5k-4,... and this way we arrange a row of 15 boys followed by 5 girls. In total its (15k)!(5k)!. Now we can divide by 15! and 5! to get rid of the inner ordering between boys and girls respectively, and then divide by k! to get rid of between the groups. But how do I go from here - or in a completely different way - and make order in each group between the 15 boys and girls.
For the first group:
$C^{15}_{15k}\times C^{5}_{5k}$ (where $C$ indicates combination)
I think this is as you did.
For the second:
$C^{15}_{15(k-1)}\times C^{5}_{5(k-1)}$
...
For the last:
$C^{15}_{15(k-(k-1))}\times C^{5}_{5(k-(k-1))}=C^{15}_{15}\times C^{5}_{5}=1$
Then, as you said, divide by $k!$ because the ordering of the groups don't matter.
So, you have
$\dfrac{1}{k!}\prod_{i=0}^{k-1}C^{15}_{15(k-i)}\times C^{5}_{5(k-i)}$ manners to get this groups...
What do you think?