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Chapter 9: Problem 64

The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) \(3,(b) 8,(c) 12,\) and \((d) 20\).

Short Answer

Expert verified
The numbers of different two-person relationships in groups of people of sizes (a) 3,(b) 8,(c) 12, and (d) 20 are 3, 28, 66, and 190 respectively.

Step by step solution

01

Determine the number of two-person relationships for group of 3

Use the formula for combinations: \( C(3,2) \). This equates to \( 3!/(2!(3-2)!) = 3 \)
02

Determine the number of two-person relationships for group of 8

Use the formula for combinations: \( C(8,2) \). This equates to \( 8!/(2!(8-2)!) = 28 \)
03

Determine the number of two-person relationships for group of 12

Use the formula for combinations: \( C(12,2) \). This equates to \( 12!/(2!(12-2)!) = 66 \)
04

Determine the number of two-person relationships for group of 20

Use the formula for combinations: \( C(20,2) \). This equates to \( 20!/(2!(20-2)!) = 190 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Combinatorics
Combinatorics is a branch of mathematics focused on counting, arranging, and grouping items. In this context, it's about finding how many ways we can create two-person relationships within a group. You might think of this like picking pairs for a dance. The basic idea is to choose a smaller subset from a larger set.
Combinatorics helps us solve problems where order doesn't matter and we want to avoid duplicating combinations. Imagine you're in a room with several people, and you want to know how many unique pairs you can form. This is where combinations come into play, using a specific formula to count these possibilities without repetition.
The Role of Factorials
Factorials are an integral part of calculating combinations. A factorial, represented by the symbol "!", means multiplying a series of descending natural numbers. For instance, 3! (read as "three factorial") equals 3 x 2 x 1 = 6.
  • This concept is critical when using combination formulas, as it helps calculate the total number of arrangements.
  • In our problem, the factorial is used to find the total number of ways to arrange the people before selecting specific combinations.
Understanding how factorials work is vital, as they help in reducing larger problems into manageable numbers. It transforms complex counting into a structured process, making it easier to compute combinations.
Exploring the Binomial Coefficient
The binomial coefficient is a key element in combinatorics, often written as \( C(n, r) \) or \( \binom{n}{r} \). It represents the number of ways to choose "r" items from a set of "n" items without regard to order.
This is exactly what you need to determine the number of two-person relationships:
  • Use the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
  • Here, \(n\) is the total number of people, and \(r\) is 2, since we're forming pairs.
By applying this formula, you can calculate the number of possible pairs in any group size. Each step involves substituting different values of \(n\) into the equation, simplifying factorial expressions, and ultimately finding the solution to the problem at hand.

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