Problem 54 - Algebra 2 and Trigonometry

Question

Rafael is running for mayor of his town. He sent out a survey asking his constituents to rank the following issues in order of importance to them: crime, unemployment, air quality, schools, public transportation, parking, and taxes. a. How many different rankings are possible? b. What fraction of the possible rankings put crime first? c. How many of the possible rankings are in alphabetical order?

Step-by-Step Solution

Answer

a. 5040 rankings. b. 1/7 of rankings have crime first. c. 1 ranking is alphabetical.

Step 1: Calculate Total Number of Rankings

To find the total number of rankings possible, we need to consider that the constituents can rank the 7 issues in any order. This is a permutation problem because the order matters. The formula for permutations of n distinct objects is \( n! \), where \( ! \) denotes a factorial. Therefore, the total number of different rankings is \( 7! \). Calculate it: \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).

Step 2: Calculate Fraction of Rankings with Crime First

To find out what fraction of rankings has crime ranked first, we need only consider the relative order of the remaining 6 issues. Once 'crime' is at the top, the other 6 issues can be arranged in any order. This number is \( 6! \), because we have 6 remaining positions. Calculate it: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). The fraction of rankings with crime first is therefore \( \frac{720}{5040} \). Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor, 720: \( \frac{720}{5040} = \frac{1}{7} \).

Step 3: Rankings in Alphabetical Order

Only one ranking is possible where the issues are ranked alphabetically: 'air quality, crime, parking, public transportation, schools, taxes, unemployment'. There are no other ways to order them alphabetically, so there is only 1 such ranking.

Key Concepts

Factorial

A factorial is a fundamental concept in combinatorics, often used to calculate permutations and combinations. It is denoted by the symbol \( ! \). The factorial of a positive integer \( n \) is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials help us determine the number of ways to arrange a set number of objects.

• The basic formula for calculating a factorial is: \( n! = n \times (n-1) \times \ldots \times 2 \times 1 \).
• Factorials grow very quickly with larger numbers.

Using factorials allows us to compute permutable configurations, such as the total number of ways constituents can rank 7 issues: \( 7! = 5040 \). This understanding is crucial for solving permutation problems in algebra.

Ranking Problems

Ranking problems in algebra involve arranging a set of items in a specific order, paying particular attention to the sequence. In the context of Rafael's mayoral survey, we look at how constituents can order issues based on importance. This is a classic permutation problem.

• Order matters in ranking problems, distinct from mere combinations where order is irrelevant.
• Each unique ordering is considered a separate result, contributing to the overall count of rankings.

When analyzing the rankings, placing 'crime' in the first position, for instance, allows us to count only the permutations of the remaining issues, illustrating how ranking problems break down into smaller, more manageable permutations.

Combinatorial Counting

Combinatorial counting is the mathematics of counting, crucial for determining the number of possible permutations and combinations in a given problem. It involves recognizing patterns and applying formulas to count arrangements effectively.

• Without combinatorial counting, determining total rankings from a large set of data would be inefficient and error-prone.
• By understanding and applying the rules of permutations and combinations, we can solve complex problems, like finding the number of ways to rank issues in Rafael’s survey.
• For instance, we determine the number of possible rankings by applying \( 7! \) for all issues or \( 6! \) when focusing on rankings with 'crime' first.

Combinatorial counting simplifies the process, reducing complex arrangements to calculations using factorials and simplifying fractions like in our survey exercise.

Algebraic Problem Solving

Algebraic problem solving involves using mathematical symbols and procedures to find solutions to various problems. It entails formulating problems in terms of algebraic expressions or equations and then finding solutions using logical reasoning.

• In Rafael's survey problem, algebra helps us formalize and simplify the process of calculating rankings.
• The step of solving for the fraction of rankings that place 'crime' first uses algebra to simplify the expression to \( \frac{1}{7} \).
• Algebra also assists in determining that there is only one alphabetical arrangement of the issues.

By using algebraic methods, we can break down seemingly complex problems into simpler, logical steps, providing a clearer path to the solution.