Problem 18 - Algebra for College Students

Question

In a seven-team baseball league, in how many ways can the top three positions in the final standings be filled? 210

Step-by-Step Solution

Answer

There are 210 ways to fill the top three positions in the league.

Step 1: Understanding the Problem

We are asked to figure out how many different ways the top 3 positions in a 7-team league can be ranked. This is essentially a permutation problem, as the order in which the teams are ranked matters.

Step 2: Setting Up the Permutation Formula

The permutation formula for selecting and arranging 'r' items out of 'n' is given by: \[ P(n, r) = \frac{n!}{(n-r)!}\].Here, we have 7 teams and we want to rank the top 3 positions, so 'n' is 7 and 'r' is 3.

Step 3: Calculating the Factorial of 'n'

Calculate the factorial of 'n', which is 7:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \].

Step 4: Calculating the Factorial of 'n-r'

Calculate the factorial of 'n-r', which is 4 (since 7 - 3 = 4):\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \].

Step 5: Using the Permutation Formula

Substitute the values we calculated into the permutation formula:\[ P(7, 3) = \frac{7!}{(7-3)!} = \frac{5040}{24} \].

Step 6: Final Calculation

Perform the division to find the number of permutations:\[ \frac{5040}{24} = 210 \].

Key Concepts

Factorials

In the realm of mathematics, especially in permutations and combinatorics, factorials play a crucial role. A factorial is represented by an exclamation mark "!" and indicates the product of all positive integers up to a given number. For example, the factorial of 7 is written as 7! and is calculated as follows:
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

Factorials are essential in permutations because they allow us to compute the total number of ways to arrange a set of items. Consider a simple set of numbers; the factorial helps determine all possible ways these numbers can be ordered. Factorials grow very quickly, which is why understanding their calculation is vital in permutation and other mathematical operations.

Permutation Formula

In combinatorics, when the order of arrangement matters, we use permutations. The permutation formula helps us calculate how to arrange 'r' items from a set of 'n' distinct items. The formula is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]

In this formula, \(n!\) is the factorial of the total number of items, and \((n-r)!\) is the factorial of the number of items left out. We use this formula when we care about the order of arrangement of the selected items. For example, if you want to find the number of ways to arrange the top 3 teams out of 7, you assign values: \( n = 7 \) and \( r = 3 \), then calculate as follows:

\[ P(7, 3) = \frac{7!}{(7-3)!} = \frac{5040}{24} = 210 \]

This indicates there are 210 different possible ways to rank the top three teams.

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting, arranging, and combination of objects. It helps us solve problems related to finite discrete structures and often involves permutations and combinations. In the context of permutations, combinatorics helps determine the number of possible arrangements.

Key concepts in combinatorics include:

• Permutations: Counting the arrangements where the order matters.
• Combinations: Counting selections where order does not matter.
• Factorials: Used extensively in calculations of permutations and combinations.

When dealing with rankings or selections where order is important, like the baseball league problem, permutations and factorials are tools of choice within combinatorics. They enable clear solutions by providing formulas and methodologies for efficient calculation.