Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Selecting theater seats Three married couples have purchased tickets for a play. Spouses are to be seated next to each other, and the six seats are in a row. In how many ways can the six people be seated?

Step-by-Step Solution

Answer

There are 48 ways to seat the six people.

Step 1: Treat Each Couple as a Unit

Since each couple needs to be seated together, treat each couple as one single unit. This changes the problem from seating 6 individuals to seating 3 units.

Step 2: Calculate Arrangements of Couples

Calculate how many ways the couples can be arranged as units. There are 3 couples, so we can arrange them in: \[ 3! = 6 \] ways.

Step 3: Calculate Arrangements Within Couples

Each couple can switch places with each other within their unit. For each couple, there are 2 ways to arrange the two people:\[ 2! = 2 \] Since there are 3 couples, this becomes: \[ 2 \times 2 \times 2 = 2^3 = 8 \] ways.

Step 4: Combine Both Results

To find the total number of ways to seat the 6 people while keeping couples together, multiply the number of ways to arrange the couples by the number of ways to arrange within each individual couple:\[ 6 \times 8 = 48 \] ways.

Key Concepts

Factorial

A factorial, denoted by the symbol "!", is a mathematical operation used in permutations and combinations to calculate the number of ways a set of objects can be arranged. The factorial of a number \( n \), written as \( n! \), is the product of all positive integers up to \( n \). For example, \( 3! = 3 \times 2 \times 1 = 6 \). Factorials are fundamental in understanding permutations, where the arrangement of a set matters.

In our theater seating problem, we use the factorial to determine how the three couples (considered as single units) can be seated. By treating each couple as a unit, we reduce the problem to arranging three units, which is handled by calculating \( 3! \).

Factorials grow quite rapidly with larger numbers, making them useful in counting problems of varying sizes, especially in discrete structures where specific arrangements are required.

Combinatorial Counting

Combinatorial counting is a principle in mathematics used to count or list items in sets with restrictions. It involves the application of combinations and permutations to figure out the different ways to arrange items. Permutations are used when the order matters, whereas combinations are used when the order does not matter.

In the case of seating arrangements, we are dealing with permutations since the order in which couples and individuals are seated is important. We begin by considering each couple as a single unit, allowing us to use permutations to calculate the seating arrangements for the units. Then, within each couple, a permutation of the two individuals bolsters the overall count.

Combinatorial counting is widely used in various fields such as computer science, statistics, and operations research to solve problems involving complex arrangements, making it an essential tool in discrete mathematics.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally distinct and separate, as opposed to continuous structures. It involves topics such as graph theory, logic, and combinatorics, which includes combinatorial counting and permutations.

In discrete mathematics, we often deal with finite sets, just like the seating arrangement problem where we have a finite number of specific chairs and people. This branch of mathematics helps solve real-world problems involving structured arrangements, like computer algorithms and coding theory.

Understanding discrete mathematics is key to effectively solving problems when we can count, list, and compare finite elements. With permutations, our current problem shows how discrete mathematics lets us find precise solutions by organizing data in structured ways. It is foundational for developing computer programs where we need to process discrete pieces of information systematically.