65. Baseball Teams A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed?

To solve the problem of how many different teams of 9 players can be formed from a baseball team of 15 members (4 pitchers and 11 other players), we can break it down into cases based on the number of pitchers included in the team. ### Step 1: Define the Cases We can have the following cases based on the number of pitchers selected for the team of 9 players: 1. **Case 1:** 0 pitchers 2. **Case 2:** 1 pitcher 3. **Case 3:** 2 pitchers 4. **Case 4:** 3 pitchers 5. **Case 5:** 4 pitchers ### Step 2: Calculate Each Case 1. **Case 1:** 0 pitchers - We choose all 9 players from the 11 non-pitchers. - The number of ways to choose 9 from 11 is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \). \[ C(11, 9) = C(11, 2) = \frac{11!}{2! \cdot 9!} = \frac{11 \cdot 10}{2 \cdot 1} = 55 \] 2. **Case 2:** 1 pitcher - We choose 1 pitcher from 4 and 8 non-pitchers from 11. \[ C(4, 1) \cdot C(11, 8) = 4 \cdot C(11, 3) = 4 \cdot \frac{11!}{3! \cdot 8!} = 4 \cdot \frac{11 \cdot 10 \cdot 9}{3 \cdot 2 \cdot 1} = 4 \cdot 165 = 660 \] 3. **Case 3:** 2 pitchers - We choose 2 pitchers from 4 and 7 non-pitchers from 11. \[ C(4, 2) \cdot C(11, 7) = 6 \cdot C(11, 4) = 6 \cdot \frac{11!}{4! \cdot 7!} = 6 \cdot \frac{11 \cdot 10 \cdot 9 \cdot 8}{4 \cdot 3 \cdot 2 \cdot 1} = 6 \cdot 330 = 1980 \] 4. **Case 4:** 3 pitchers - We choose 3 pitchers from 4 and 6 non-pitchers from 11. \[ C(4, 3) \cdot C(11, 6) = 4 \cdot C(11, 5) = 4 \cdot \frac{11!}{5! \cdot 6!} = 4 \cdot \frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 4 \cdot 462 = 1848 \] 5. **Case 5:** 4 pitchers - We choose all 4 pitchers and 5 non-pitchers from 11. \[ C(4, 4) \cdot C(11, 5) = 1 \cdot C(11, 5) = 1 \cdot 462 = 462 \] ### Step 3: Sum All Cases Now, we sum the number of teams from all cases: \[ \text{Total} = 55 + 660 + 1980 + 1848 + 462 \] ### Step 4: Calculate the Total Let's calculate the total: \[ \text{Total} = 55 + 660 + 1980 + 1848 + 462 = 4005 \] Thus, the total number of different teams of 9 players that can be formed is \( \boxed{4005} \).