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$\(^n_k\)$ is the number of ways to choose $k$ team members from $n$ players.

$k$ is the number of ways to choose one captain from $k$ team members. (It isn't the number of ways to choose a captain from the original group of people, as you suggested; that would be $n$.)

$k\(^n_k\)$ is the product of the two, which is the number of ways to do both independently.