Problem 70 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for each of the following arrangements? (a) The women are to be seated together. (b) The men and women are to be seated alternately by gender.

Step-by-Step Solution

Answer

(a) 2880 ways; (b) 576 ways.

Step 1: Women Seated Together as a Single Unit

First, treat the four women as a single unit or block. This block, along with the four individual men, forms a total of five items to arrange. Calculate the number of ways to arrange these 5 items:\[5! = 120\]

Step 2: Arrange Women Within the Block

Next, determine the number of ways to arrange the four women within their block. Since they can be rearranged among themselves:\[4! = 24\]

Step 3: Total Arrangements for Part (a)

To find the total number of seating arrangements where the women are seated together, multiply the results of Step 1 and Step 2:\[5! \times 4! = 120 \times 24 = 2880\]

Step 4: Alternate Seating Arrangement for Part (b)

The alternate condition specifies that the sequence should be MWMWMWMW or WMWMWMWM, where M represents men and W represents women. Choose one pattern (e.g., MWMWMWMW), and calculate the arrangements for each group separately:For men and women, each group has 4 members. So there are:\[4! \times 4! = 24 \times 24 = 576\]

Key Concepts

Combinatorics

Combinatorics is the mathematics of counting, arranging, and combining objects. It helps us to determine the number of ways in which certain tasks can be completed. When you think about placing people in seats, or arranging various items, combinatorics is at play. In the specific context of seating arrangements, combinatorics helps us explore all possible configurations effectively.

When dealing with arrangements, we often consider permutations, where the order of arrangement is key. For instance, swapping the places of two people results in a different permutation even though the same individuals are involved. This principle is crucial in solving many problems involving sequences and ordering.

• **Permutations:** Arrangements where order matters.
• **Combinations:** Selections where order doesn't matter.
• **Blocks or Units:** Assumed groups within larger permutations to simplify calculations.

By reducing complex scenarios into these straightforward principles, solving large arrangement problems becomes more manageable through combinatorics.

Factorial

The factorial, denoted by the exclamation mark (!), is a foundational concept in combinatorics. It represents the product of all positive integers up to a certain number. For example, factorial of 5, written as 5!, equals 5 x 4 x 3 x 2 x 1 = 120. Factorials are used to calculate permutations, which describe different ways to arrange items.

Factorials grow incredibly fast with larger numbers, turning what seems like countless possibilities into a manageable count of arrangements. This is why factorial is crucial when dealing with arrangements of any kind, especially seating.

• **Notation:** The symbol '!'
• **Calculation:** Involves multiplying sequential numbers
• **Importance:** Essential for counting permutations

In seating arrangements, like in our exercise, factorial yields the number of ways to position each individual or group unit within a sequence.

Seating Arrangements

Seating arrangements focus on positioning individuals in a line, around a table, or in any seating pattern. The order in which each person sits defines the uniqueness of an arrangement. In specific exercises, like our example, specific constraints are often applied, which direct how individuals can be seated.

The exercise given is a perfect illustration. There are varied ways to think about fitting people into seats:

• **Single Unit or Block Assumptions:** Treating a group of people as a single unit for simplicity, like seating four women together.
• **Alternating Patterns:** Ensuring arrangements like alternate male and female seating, where specific sequences must be followed.
• **Calculation:** Often involves both direct permutations of the whole group and internal arrangements within assumed blocks.

By breaking down a problem into these understandable pieces, arranging people into seats becomes less overwhelming, offering clear paths to precise solutions.