Step 1: We can think of this selection as splitting the 10 players into three separate groups at once: 1 player becomes the captain, 4 more players join as the rest of the team, and the remaining 5 players are not selected at all.
Step 2: The number of ways to split a group of 10 into groups of sizes 1, 4, and 5 is given by the multinomial coefficient: \( \dfrac{10!}{1!\,4!\,5!} \).
Step 3: Compute the factorials and simplify: \(10! = 3628800\), \(1!=1\), \(4!=24\), \(5!=120\), so the denominator is \(1 \times 24 \times 120 = 2880\). Dividing gives \( \dfrac{3628800}{2880} = 1260 \).
\[ \boxed{1260} \]