A certain professional soccer team runs a 4-4-2 formation on the field. This means they play with 4 defenders, 4 midfielders, 2 forwards, and 1 goalkeeper on the field. If the team has 6 defenders, 5 midfielders, 3 forwards, and 2 goalkeepers, how many different groups of 11 starting players could they have?

note that:Each player only plays in his designated position. The 2nd goalkeeper is not allowed to play as a defender.

To find the number of different groups of 11 starting players, we need to consider the restrictions given. Let's break it down step by step.

1. We have 6 defenders, but only 4 are needed on the field. So, we need to choose 4 defenders from the available 6 defenders. This can be done using combinations or "n choose k" formula, denoted as C(n,k), which calculates the number of ways to choose k items from a set of n items without considering their order. In this case, we want to choose 4 defenders from 6, so we can calculate it as C(6,4).

C(6,4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4!)/(4! * 2 * 1) = (6 * 5) / (2 * 1) = 15

Therefore, there are 15 different ways to choose 4 defenders from the available 6 defenders.

2. We have 5 midfielders, but only 4 are needed on the field. Similarly, we need to choose 4 midfielders from the available 5 midfielders. Using the same logic, we can calculate it as C(5,4).

C(5,4) = 5! / (4!(5-4)!) = 5! / (4!1!) = 5

Therefore, there are 5 different ways to choose 4 midfielders from the available 5 midfielders.

3. We have 3 forwards, but only 2 are needed on the field. Again, we need to choose 2 forwards from the available 3 forwards. Using the same logic, we can calculate it as C(3,2).

C(3,2) = 3! / (2!(3-2)!) = 3! / (2!1!) = 3

Therefore, there are 3 different ways to choose 2 forwards from the available 3 forwards.

4. We have 2 goalkeepers, and only 1 is needed on the field. So, we don't need to make any choices here.

Now, to find the total number of different groups of 11 starting players, we need to multiply the number of choices for each position together.

Total number of different groups = C(6,4) * C(5,4) * C(3,2) * 1

Total number of different groups = 15 * 5 * 3 * 1

Total number of different groups = 225

Therefore, the professional soccer team could have 225 different groups of 11 starting players given the specified restrictions.