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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 $({10 \choose 2}*{8 \choose 2}*{6 \choose 2}*{4 \choose 2}*{2 \choose 2})/5!$

  • 0
    why have you divided by 5! – 2017-02-11
  • 0
    I divided by 5! to correct for overcounting – 2017-02-11
  • 0
    where have you done overcounting?? – 2017-02-11

2 Answers 2

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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

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You have done a mistake. It will be:

$${10 \choose 2}*{8 \choose 2}*{6 \choose 2}*{4 \choose 2}$$