63. Baseball In the American Baseball League, a designated hitter may be used. How many batting orders is it possible for a manager to use? (There are 9 regular players on a team.)

To determine how many batting orders are possible with a designated hitter in a baseball team with 9 regular players, we can follow these steps: 1. **Understanding the Setup**: - There are 9 regular players. - One of these players will be replaced by a designated hitter (DH), making it effectively 10 players to choose from for the batting order. 2. **Choosing the Designated Hitter**: - The manager can choose any one of the 9 regular players to be the designated hitter. 3. **Calculating the Batting Order**: - Once the designated hitter is chosen, there are 9 players (the 8 remaining regular players plus the designated hitter) to arrange in the batting order. - The number of ways to arrange \( n \) players is given by \( n! \) (n factorial). 4. **Calculating the Total Orders**: - The total number of batting orders can be calculated as: \[ \text{Total Orders} = \text{Number of Choices for DH} \times \text{Arrangements of Players} \] - This can be expressed mathematically as: \[ \text{Total Orders} = 9 \times 9! \] Now, let's calculate \( 9! \) and then the total number of batting orders. First, we calculate \( 9! \): \[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] Now, I will calculate \( 9! \) and then multiply it by 9. Calculate the value by following steps: - step0: Calculate: \(9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\) - step1: Rewrite the expression: \(9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\) - step2: Multiply the terms: \(72\times 7\times 6\times 5\times 4\times 3\times 2\) - step3: Multiply the terms: \(504\times 6\times 5\times 4\times 3\times 2\) - step4: Multiply the terms: \(3024\times 5\times 4\times 3\times 2\) - step5: Multiply the terms: \(15120\times 4\times 3\times 2\) - step6: Multiply the terms: \(60480\times 3\times 2\) - step7: Multiply the terms: \(181440\times 2\) - step8: Multiply the numbers: \(362880\) Calculate or simplify the expression \( 9*362880 \). Calculate the value by following steps: - step0: Calculate: \(9\times 362880\) - step1: Multiply the numbers: \(3265920\) The calculation yields: 1. \( 9! = 362880 \) 2. The total number of batting orders is: \[ 9 \times 9! = 9 \times 362880 = 3265920 \] Thus, the total number of possible batting orders with a designated hitter in the American Baseball League is **3,265,920**.