Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The different two-person relationships possible in groups of sizes 3, 8, 12, and 20 are 3, 28, 66, and 190 respectively.

Step 1: Compute for size 3

Plug \(n = 3\) and \(r = 2\) into the formula \(C(n, r) = n! / [r!(n - r)!]\). So we have \(C(3, 2) = 3! / [2!(3 - 2)!] = 3\). Thus, there are 3 different two-person relationships possible in a group of size 3.

Step 2: Compute for size 8

Plug \(n = 8\) and \(r = 2\) into the formula \(C(n, r) = n! / [r!(n - r)!]\). So we have \(C(8, 2) = 8! / [2!(8 - 2)!] = 28\). Thus, there are 28 different two-person relationships possible in a group of size 8.

Step 3: Compute for size 12

Plug \(n = 12\) and \(r = 2\) into the formula \(C(n, r) = n! / [r!(n - r)!]\). So we have \(C(12, 2) = 12! / [2!(12 - 2)!] = 66\). Thus, there are 66 different two-person relationships possible in a group of size 12.

Step 4: Compute for size 20

Plug \(n = 20\) and \(r = 2\) into the formula \(C(n, r) = n! / [r!(n - r)!]\). So we have \(C(20, 2) = 20! / [2!(20 - 2)!] = 190\). Thus, there are 190 different two-person relationships possible in a group of size 20.

Key Concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics. They help us understand how to count arrangements and selections in a set. Permutations are used when the order of arrangement matters. Combinations, on the other hand, are employed when the order does not matter, which is exactly what we need for choosing two-person relationships from a group.

The formula for combinations is used when you want to count how many ways you can choose a subset of size \(r\) from a larger set of size \(n\). This is given by the formula \[ C(n, r) = \frac{n!}{r!(n - r)!} \] where \(n!\) denotes the factorial of \(n\).

In the context of two-person relationships, \(r = 2\) because we are forming pairs, and \(n\) is the number of people in the group. This simplification leads to the formula \(C(n, 2)\) which helps easily determine the number of unique pairs possible in a group.

Factorials in Mathematics

Factorials are a key component in permutations and combinations. A factorial, denoted as \(n!\), is the product of all positive integers up to \(n\). For example:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

This concept helps count the total number of distinct arrangements possible for \(n\) items. Factorials grow very quickly with increasing \(n\), which is why understanding their properties is crucial in combinatorics.

In combinations, factorials appear in both the numerator and the denominator. They ensure that the order of selection does not matter by dividing out duplicate arrangements. Comprehending how factorials reduce the complexity of counting combinations is essential for solving problems involving them.

Two-person relationships in groups

When analyzing groups, we often want to understand the potential connections between members. Two-person relationships are an interesting case as they describe basic pairwise interactions. The problem of determining the number of such relationships in a group of size \(n\) translates to finding how many unique pairs can be formed.

Using the combinations formula \(C(n, 2)\), we can easily calculate the number of two-person relationships without considering the order. This problem can be visualized on simpler sets to grasp the concept more intuitively. For example, if there are 3 people, namely A, B, and C, the potential pairs are (A, B), (A, C), and (B, C).

The concept scales with group size, and as demonstrated in the examples of the step-by-step solution, it reveals an increasing complexity as the group size grows. This illustrates the exponential growth of relationships and their significance in social dynamics. Mastering this helps solve not just mathematical problems but also entails deeper social and organizational analysis.