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In an international track competition, there are five U.S. athletes, four Russian athletes, three French athletes, and one German athlete. How many rankings of the $13$ athletes are there when:

(a) only nationality is counted?: $\displaystyle\frac{13!}{5!4!3!1!}=360,360$ rankings

(b) This is what I need help with: Only nationality is considered and all the U.S. athletes finish before the Russian athletes?

Only nationality considered comes from part (a).

My approach is to consider the athletes as letters:

UUUUU RRRR FFF G

There are $\displaystyle\frac{13!}{5!4!}$ ways to arrange the U's and R's. There is $1$ way in which the U.S. athletes finish before the Russian athletes. I am stuck here. How should this part be approached?

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  • $\begingroup$ Suppose you know the relative order in which the US and Russian athletes finish. What about the remaining four? $\endgroup$
    – user418131
    Oct 2 '18 at 18:14
  • $\begingroup$ Why is there only one way in which US athletes finish before the Russian athletes? In the configuration you stated, the $G$s and $F$s could very well be in the middle thus giving at the very least an additional possible outcome. $\endgroup$
    – qualcuno
    Oct 2 '18 at 18:17
  • $\begingroup$ this is only my approach..i didnt say my approach was right...if it is wrong please tell me where i am wrong...@GuidoA. $\endgroup$
    – rover2
    Oct 2 '18 at 18:20
  • $\begingroup$ I was just trying to understand if some parts of the problem were missing :) I'm thinking about this at the moment, I'll try to come up with an answer. $\endgroup$
    – qualcuno
    Oct 2 '18 at 18:20
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    $\begingroup$ @AnotherJohnDoe can the remaining four be ranked in $\displaystyle\frac{4!}{3!1!}$ ways? $\endgroup$
    – rover2
    Oct 2 '18 at 18:21
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There are $\binom{13}{3}$ ways to choose the positions of the French athletes and $\binom{10}{1}$ ways to choose one of the remaining ten positions for the German athlete. That leaves nine positions to fill with athletes from the United States and Russia. There is only one way to fill these positions so that every American athlete comes before every Russian athlete. Hence, the number of ways all the American athletes can finish ahead of all the Russian athletes if only nationalities are considered is $$\binom{13}{3}\binom{10}{1}$$

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  • $\begingroup$ You are geniuous man! $\endgroup$
    – Aqua
    Oct 2 '18 at 19:28

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