Maths challenge problem: Why is the number of teams which require 4 substitutions 32?

Question

I came across the following problem on a UKMT senior maths challenege:

A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 defender, 1 midfielder and 1 forward. A substitute may only replace a player in the same category e.g. midfielder for midfielder. Given that a maximum of 3 substitutes may be used and that there are still 11 players on the pitch at the end, how many different teams could finish the game?

Here is the official solution:
Firstly, we note that of the players on the pitch at the end of the game, the goalkeeper is 1 of 2 players, the 4 defenders form 1 of 5 different possible combinations, as do the 4 midfielders, and the 2 forwards form 1 of 3 different possible combinations. So if up to 4 substitutes were allowed, the number of different teams which could finish the game would be $2\times 5\times5\times3$, that is 150. From this number we must subtract the number of these teams which require 4 substitutions to be made. This is $1\times 4\times4\times2$, that is 32, so the required number of teams is 118.

When attempting the problem, I got as far as $2\times 5\times5\times3$, that is 150, and knew that the number of these teams which require 4 substitutions was needed, but why is the number of teams which require 4 substitutions to be made $1\times 4\times4\times2=32$? Thanks!

Answer 1

The problem states that "Given that a maximum of 3 substitutes may be used".

This means that from your $2*5*5*3$ you need to subtract the number of possible teams with all 4 substitutes used. If all 4 substitutes are used, there are 4 ways to choose the other 3 defenders out of 4 non-substitutes, 4 ways to choose the other 3 midfielders out of the 4 non-substitutes, and 2 ways to choose the other forward out of 2 non-substitutes. Using the multiplication principle, this gives $1*4*4*2=32$ as the total number of ways to choose a team with all 4 substitutes being used.

Now subtract that number from $150$ to get the total number of possible teams without having all 4 substitutes being used.