# Trouble understanding the answer to combinations question

combinationscombinatorics

I attempted the following question

How many ways can a team of 24 hockey players choose a captain and two
alternate captains?


The correct answer in the end turned out to be 6072, computed by multiplying 24 by the result of the combination $$C(23, 2) = 253$$:

24 * $$\dfrac{23!}{2!(23-2)!}$$

I do not understand why this is the case however. Why make the distinction between the 24 and then the combination of $$C(23, 2)$$? My initial attempt involved simply using the combination formula with with 24 as the total and 3 as the sample, that being the captain and 2 alternate captains in the problem. This would have given me:

$$\dfrac{24!}{3!(24-3)!} = 2024$$

I am struggling to figure out what I am overlooking in this problem.

After having chosen $$3$$ special players from the $$24$$ players (via one of $$\displaystyle\binom{24}3$$ possible ways),

you next need to elect one of the three to be the actual captain (via one of $$3$$ possible ways).