In a college team there are 15 players of whoom 3 are teachers . In how many ways can a team of 11 players be selected so as to include
at least one teachers ?

Answer

To solve the problem of selecting a team of 11 players from a group of 15 players (including 3 teachers), ensuring that at least one teacher is included, we can use the principle of combinations. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the total players and their categories
We have a total of 15 players:
- 3 teachers
- 12 non-teachers

### Step 2: Calculate the total ways to select 11 players without restrictions
The total number of ways to select 11 players from 15 players is given by the combination formula:
\[ \text{Total ways} = \binom{15}{11} \]
This can also be calculated as:
\[ \binom{15}{11} = \binom{15}{4} \quad \text{(since } \binom{n}{r} = \binom{n}{n-r}\text{)} \]
Calculating this:
\[ \binom{15}{4} = \frac{15!}{4!(15-4)!} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365 \]

### Step 3: Calculate the ways to select 11 players with no teachers
Next, we calculate the number of ways to select 11 players without any teachers. This means selecting all players from the 12 non-teachers:
\[ \text{Ways without teachers} = \binom{12}{11} = 12 \]

### Step 4: Calculate the ways to select 11 players with at least one teacher
To find the number of ways to select a team of 11 players that includes at least one teacher, we subtract the number of ways to select 11 players without any teachers from the total ways:
\[ \text{Ways with at least one teacher} = \text{Total ways} - \text{Ways without teachers} \]
Substituting the values we calculated:
\[ \text{Ways with at least one teacher} = 1365 - 12 = 1353 \]

### Final Answer
Thus, the total number of ways to select a team of 11 players that includes at least one teacher is:
\[ \boxed{1353} \]