Question

Three friends are going to the theater. They arrive just in time to get the last 3 seats that are available together in a row. How many different seating arrangements can the friends sit in?

Short answer

There are \(\textbf{6}\) different seating arrangements for the three friends in a row.

Step-by-step solution

Understand the concept of permutation

Permutation refers to the number of ways a set of objects can be arranged. In this context, each arrangement of friends is different; that is, each of the seats can be filled by any friend. In general, when we have 'n' distinct items, they can be arranged in 'n!' ways, where 'n!' refers to the product of all positive integers up to 'n'.

Identify the number of objects to arrange

In this problem, the objects to arrange are the three friends. Therefore, we have n = 3.

Calculate the number of permutations

To calculate the number of permutations, do the factorial of the number of objects. Mathematically, you can represent the factorial by using the '!' symbol. So, we will calculate the factorial of 3 (which is written as 3!) as follows: 3! = 3 × 2 × 1 = 6.

Interpret the result

The calculated result, 6, tells us that there are 6 different possible seating arrangements between the three friends.

Key concepts

Factorials

Factorials are a foundational concept in permutations and combinatorics. The idea of a factorial is simple: it's the product of all positive numbers up to a given number. For example, if we have the number 3, the factorial (written as 3!) is calculated as follows:
\[ 3! = 3 \times 2 \times 1 = 6 \]
Factorials grow rapidly as the number increases, becoming larger quickly. This notion is crucial in determining the amount of different arrangements possible for a set of objects. In the context of our theater seat problem, knowing how to compute a factorial allowed us to ascertain all the seating possibilities.

• The factorial of 0 is defined as 1 (0! = 1)
• Factorials apply only to whole numbers
• This mathematical tool is used extensively in probability and statistics

When arranging items, factorials help us find out how things can be sorted or placed uniquely.

Arrangements

Arrangements highlight the various ways objects or people can be sequenced or placed. In our scenario, the three friends sitting in three seats can be arranged in multiple ways. But what exactly does this mean?
For permutations, each position matters. Switching positions creates a new unique arrangement. For instance, if Friend A sits next to Friend B, with Friend C on the last seat, it is considered a different arrangement than if Friend A swaps spots with Friend C.

• Order is significant; changing positions creates a new arrangement
• Each arrangement is distinct, especially when dealing with unique entities (like people)
• The concept applies universally to arranging numbers, letters, and other items

In broader combinatorics, arrangements are studied to understand possibilities, optimize arrangements, and solve complex problems.

Combinatorics

Combinatorics is an extensive field of mathematics focused on counting, arranging, and analyzing finite sets of elements. It embraces permutations, combinations, and various other concepts.
In simpler terms, combinatorics helps you determine how many ways you can place a set of items. With permutations, it's about arranging distinct objects. Combinatorics allows for counting possibilities in an organized way.

• Permutations (order matters) and combinations (order doesn't matter) are key areas
• Combinatorial principles are crucial in probability theory and various optimization problems
• Real-life applications include seating guests, scheduling, and resource allocation

Understanding these principles aids in making informed decisions on how elements can be organized, moving beyond just theoretical exercises to practical applications.