Problem 49 - Precalculus with Limits

Question

A baseball coach is creating a nine-player batting order by selecting from a team of \( 15 \) players. How many different batting orders are possible?

Step-by-Step Solution

Answer

The number of different batting orders possible is \( 15! / 6! \)

Step 1: Understanding the problem

We're given a total of \( 15 \) players and we need to select \( 9 \) players for a batting order. The fact that the order of players matters makes this a permutation problem. To calculate this we can use the permutation formula \( nPr = n! / (n - r)! \) where \( n = 15 \) (total number of players), and \( r = 9 \) (number of players in the batting order).

Step 2: Apply the permutation formula

The permutation formula is \( nPr = n! / (n - r)! \). Put \( n = 15 \) and \( r = 9 \) into the permutation formula, we have: \( 15P9 = 15! / (15 - 9)! = 15! / 6! \).

Step 3: Calculate the factorial

Use the factorial function to calculate \( 15! \) and \( 6! \). The factorial function is a product of an integer and all the integers below it. It's denoted by the exclamation mark, !. So \( 15! = 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 \) and \( 6! = 6 × 5 × 4 × 3 × 2 × 1 \). The quotient of these two factorials will give the total number of different batting orders possible.

Step 4: Calculate the number of permutations

The result of \( 15! / 6! \) will give us the number of different permutations possible with \( 15 \) players and a batting order of \( 9 \) players. This way we can find out how many different batting orders are possible.

Key Concepts

Factorials and Their Application

In mathematics, the concept of the factorial, represented by an exclamation mark (!), plays a crucial role in several calculations, especially those involving permutations and combinations. A factorial of a non-negative integer, say \( n \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It provides a way to determine the number of ways objects can be arranged in order.
Factorials are key in calculating permutations, which is why they appear in the permutation formula \( nPr = \frac{n!}{(n-r)!} \). Here, the factorial helps to account for arranging a smaller number of items (\( r \)) out of a larger set (\( n \)).

• Factorials grow rapidly: even small numbers when factorialized become very large.
• Mathematically essential: used in various fields, from statistics to economic modeling.

Counting Methods and Permutations

Counting methods are strategies used to determine the number of possible arrangements or selections of objects. One of the most common counting methods in mathematics is permutations.
Permutations are used specifically when the arrangement or order of the objects matters. For example, in a batting order for baseball, each player's position is crucial, making it a permutation problem.
The formula for permutations is \( nPr = \frac{n!}{(n - r)!} \), where:

• \( n \) is the total number of objects.
• \( r \) is the number of objects to arrange.

This formula calculates the number of possible ways to choose \( r \) objects from \( n \) objects and arrange them in order.
In our problem, selecting 9 players out of 15 and arranging them in order is a classic example of permutations, as illustrated by the solution.

Combinatorics and Its Versatility

Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and structuring sets of elements. It comes in handy in solving problems involving permutations and combinations. While permutations focus on the order, combinations focus merely on the selection of items.
Combinatorics offers systematic counting principles, such as the multiplication principle and addition principle, that help simplify complex counting scenarios.

• Permutations: Important when order matters.
• Combinations: Used when the order does not matter (not directly applied here, but good to know).

Combinatorics is applied in diverse fields such as computer science, optimizing network paths, electoral system methods, and even biological arrangements.
Understanding these concepts enriches our ability to approach and solve a variety of real-world problems through a mathematical lens.