Picking teams of 2 for a country where team members and countries are distinguishable? A country has 10 diplomats, and needs to assign two of them to each of 5 countries. How many assignments are possible? 
(The diplomats and the countries are distinguishable.)
Would the possibilities be as simple as $({10 \choose 2}*{8 \choose 2}*{6 \choose 2}*{4 \choose 2}*{2 \choose 2})/5!$

Pick any two: $\dbinom{10}{2}$
Assign to any country: $\dbinom{5}{1}$
Pick any remaining two: $\dbinom{8}{2}$
Assign to any remaining country: $\dbinom{4}{1}$
Pick any remaining two: $\dbinom{6}{2}$
Assign to any remaining country: $\dbinom{3}{1}$
Pick any remaining two: $\dbinom{4}{2}$
Assign to any remaining country: $\dbinom{2}{1}$
Total count is

$\dbinom{10}{2}\dbinom{5}{1}\dbinom{8}{2}\dbinom{4}{1}\dbinom{6}{2}\dbinom{3}{1}\dbinom{4}{2}\dbinom{2}{1}$

To avoid over counting, due to symmetry, we have to divide the above result by 

$5!$

Thus, 
Total count is

$\dbinom{10}{2}\dbinom{8}{2}\dbinom{6}{2}\dbinom{4}{2}$

This agrees with the answer given by @user8795
