Problem 73 - Algebra and Trigonometry

Question

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 first seat is to be occupied by a man. (b) The first and last seats are to be occupied by women.

Step-by-Step Solution

Answer

(a) 20160 ways; (b) 8640 ways.

Step 1: Determine arrangement requirements for part (a)

For part (a), we know that the first seat must be occupied by a man. That leaves us with 3 remaining men and 4 women to seat in the remaining 7 positions.

Step 2: Calculate the seating for part (a)

The first seat has 4 choices (since there are 4 men). For the remaining 7 seats, we distribute the remaining 3 men and 4 women. The number of ways to arrange 7 people is 7 factorial, or 7!. Therefore, the total is the product of choosing the man for the first seat and arranging the other 7: \[ 4 imes 7! = 4 imes 5040 = 20160 \] ways.

Step 3: Determine arrangement requirements for part (b)

In part (b), the first and last seats should be occupied by women. This gives us 4 choices for the first seat and 3 choices for the last seat (since one woman is already seated first). This leaves 4 men and 2 women to seat in the remaining 6 positions.

Step 4: Calculate the seating for part (b)

The first seat has 4 choices and the last seat has 3 choices for the women. For the remaining 6 positions, arrange 4 men and 2 women. The number of ways to arrange 6 people is 6!. Therefore, the total is the product of choosing women for the first and last seats and arranging the remaining 6 seats: \[ 4 imes 3 imes 6! = 4 imes 3 imes 720 = 8640 \] ways.

Key Concepts

Understanding Counting Principles

Counting principles are powerful tools used in mathematics to determine the number of possible outcomes or arrangements in various situations. When dealing with problems that involve arranging people or objects, the main principles used are the **multiplication principle** and the **addition principle**.

• The multiplication principle states that if one event can occur in \( m \) ways and another independent event can occur in \( n \) ways, then there are \( m \cdot n \) ways in which both events can occur together.
• The addition principle is used when we want to count the number of ways in which one or another mutually exclusive event can occur, simply by adding the number of ways each event can occur.

In problems like seating arrangements, these principles help us figure out how to distribute people into specific positions or satisfy given conditions systematically, especially when we have restrictions like certain seats needing to be filled by specific types of people (men or women, in this context). This systematic approach prevents mistakes and provides a clear path to the solution.

Seating Arrangements

Seating arrangements require applying counting principles to determine the number of possible ways people or objects can be positioned. This type of problem usually comes with specific constraints, which affect how calculations are carried out. To illustrate, consider this problem where we have to seat 4 men and 4 women in a row:

• In part (a), the first seat must be occupied by a man. We start by selecting one of the 4 men to take the first seat. This leaves us with 3 men and 4 women, which need to be arranged in the remaining 7 seats. The number of ways to do this is obtained by calculating the factorial of 7, since the order matters for every person.
• In part (b), both the first and last seats must be reserved for women. We select one woman for the first seat, leaving us with 3 choices for the final seat (as one woman is already seated first). With 4 men and the remaining 2 women for the middle seats, finding the arrangement requires the factorial of 6 to determine all possible orders.

These calculations demonstrate that knowing who goes where at first enables a structured allocation of the remaining individuals. Breaking it down part-by-part simplifies complex arrangements into manageable steps.

Factorials and Their Role

Factorials, denoted by \( n! \), are crucial in permutations and combinations. A factorial is the product of all positive integers up to a specified number \( n \). It symbolizes the total number of ways to arrange \( n \) distinct objects into a sequence.For instance, \( 7! \) means \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \). This number represents the many potential permutations of seven uniquely identifiable positions. In seating problems like ours, calculating \( 7! \) or \( 6! \) is crucial after accounting for initial seat assignments.

• Factorials are instrumental because they capture all possible permutations for any given set size.
• They help simplify calculations when determining how to arrange remaining participants after certain positions are predefined.

Understanding how factorials work aids us in efficiently handling permutation-related problems like seating arrangements. Hence, it’s indispensable mathematics that unlocks complexity in counting scenarios.