In how many ways can a team of 11 players be selected from 14 players when two of them can play as goalkeepers only?

Answer

To solve the problem of selecting a team of 11 players from 14 players, where 2 of them can only play as goalkeepers, we can break it down into steps.

### Step-by-Step Solution:

1. **Identify the Total Players and Goalkeepers**:
We have a total of 14 players, and out of these, 2 players can only play as goalkeepers.

2. **Select the Goalkeeper**:
Since we can only choose one goalkeeper from the two available, we can calculate the number of ways to select the goalkeeper. This can be done using the combination formula:
\[ \text{Ways to choose 1 goalkeeper from 2} = \binom{2}{1} = 2 \]

3. **Determine Remaining Players**:
After selecting one goalkeeper, we need to select the remaining players to complete the team of 11. Since we have already chosen 1 goalkeeper, we need to select 10 more players.

4. **Calculate Remaining Players**:
After selecting 1 goalkeeper, the number of players left to choose from is:
\[ 14 - 1 = 13 \text{ players (1 goalkeeper selected)} \]
However, we still have the 12 players left (14 total - 2 goalkeepers + 1 selected goalkeeper).

5. **Select 10 Players from Remaining 12**:
Now, we need to select 10 players from the remaining 12 players (who can play any position). The number of ways to select 10 players from 12 can be calculated as:
\[ \text{Ways to choose 10 players from 12} = \binom{12}{10} = \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \]

6. **Calculate Total Combinations**:
To find the total number of ways to form the team of 11 players, we multiply the number of ways to choose the goalkeeper by the number of ways to choose the remaining players:
\[ \text{Total ways} = \text{Ways to choose goalkeeper} \times \text{Ways to choose remaining players} = 2 \times 66 = 132 \]

### Final Answer:
Thus, the total number of ways to select a team of 11 players from 14 players, given the conditions, is **132 ways**.

---