Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Baseball batting order After selecting nine players for a baseball game, the manager of the team arranges the batting order so that the pitcher bats last and the best hitter bats third. In how many different ways can the remainder of the batting order be arranged?

Step-by-Step Solution

Answer

5040 ways.

Step 1: Identify Given Information

We know that there are nine players on the baseball team. The manager fixes the position of two players: the pitcher will bat last and the best hitter will bat third. This leaves us with seven positions and seven players that need to be arranged.

Step 2: Determine Players to Arrange

Since two positions are fixed (third and ninth), we need to arrange the remaining 7 players (not the pitcher or the best hitter) in the remaining 7 positions.

Step 3: Apply Permutation Formula

The number of ways to arrange 7 players in 7 positions is given by the permutation formula: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 4: Calculate Permutations

Calculating the permutation gives:\[7! = 5040\]Thus, there are 5040 different ways to arrange the remaining batsmen in the batting order.

Key Concepts

Baseball Batting Order

In baseball, the batting order refers to the sequence in which players come up to bat during a game. Each team has a lineup, usually consisting of nine players, and the order in which they bat is strategically determined by the team's manager. This order can greatly influence the flow of the game because different players have different strengths. Some may excel at hitting home runs, while others might be better at getting on base. In the exercise, the manager decided to have the pitcher bat last and the best hitter bat third. This decision fixes two positions in the batting order, and understanding how to arrange the remaining players is essential to create a strong lineup.

Arranging Players

Arranging players in a batting lineup means deciding the sequence in which the players will bat. In the given exercise, there are specific restrictions: the pitcher must bat last and the best hitter must bat third. This means those two positions are fixed. The challenge is to arrange the remaining seven players into the remaining seven positions. Since each position affects the team's overall performance, coaches must factor in players' abilities and strategies. One might agree that placing players who get on base frequently could bat earlier to maximize scoring chances, while players who hit with power might bat later to drive runners home.

Factorial Calculation

Factorial calculation is a mathematical concept used for organizing several items. In the context of this problem, it's applied to arranging the seven remaining players. The factorial of a number, represented by the symbol '!', is the product of all positive integers up to that number. For example, the factorial of 7, denoted as 7!, means:

• Multiply 7 by 6: 7 × 6 = 42
• Then, multiply the result by 5: 42 × 5 = 210
• Continue multiplying by 4, 3, 2, and finally 1

This results in 7! = 5040, meaning there are 5040 different ways to arrange the seven players. Factorial calculations are fundamental in determining permutations, which are all the possible arrangements of a set of items.

Step by Step Solution

Understanding the solution involves following a series of steps: First, identify the known conditions: two positions in the batting order are set, leaving the remaining seven players to fill seven spots. Next, using the concept of permutations, calculate how these players can be arranged. Permutation involves calculating the factorial of the number of items to arrange, which in this exercise is 7. By computing 7! = 5040, you find how many possible arrangements exist for these players. Step-by-step breakdown: - Start with the number of players needing arrangement (7 in this case). - Use the factorial formula, applying multiplication successively down to 1. This methodical approach ensures each step is clear, resulting in a robust understanding of permutations.