Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

A basketball squad consists of twelve players. (a) Disregarding positions, in how many ways can a team of five be selected? (b) If the center of a team must be selected from two specific individuals on the squad and the other four members of the team from the remaining ten players, find the number of different teams possible.

Step-by-Step Solution

Answer

(a) 792 ways, (b) 420 ways.

Step 1: Understanding and Setting Up Part (a)

We need to select 5 players from a total of 12 players. For this part, we're selecting players without regard to position, which means we're dealing with combinations.

Step 2: Calculating Combinations for Part (a)

To find the number of ways to select 5 players out of 12, we use the combination formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \( n = 12 \) and \( k = 5 \). Substituting these values, we get:\[ C(12, 5) = \frac{12!}{5! \cdot 7!} = \frac{479001600}{120 \cdot 5040} = 792 \]

Step 3: Understanding and Setting Up Part (b)

The center of the team must be selected from two specific players, while the remaining 4 players are selected from the remaining 10 players.

Step 4: Choosing the Center for Part (b)

Since the center is chosen from two individuals, there are 2 ways to choose 1 person for the center position.

Step 5: Calculation for Remaining Team Members in Part (b)

The remaining 4 players need to be chosen from 10 available players. Using the combination formula again, we calculate:\[ C(10, 4) = \frac{10!}{4! \cdot 6!} = \frac{3628800}{24 \cdot 720} = 210 \]

Step 6: Calculating Total Teams for Part (b)

Since there are 2 ways to choose the center, and for each center, there are 210 ways to choose the remaining team members, we multiply these values:\[ 2 \times 210 = 420 \]

Key Concepts

Permutations

Permutations involve the arrangement of objects in a specific order. In permutations, order matters, which differentiates them from combinations. Imagine lining up books on a shelf. The order of books matters, and this is a permutation. However, in some cases, like our basketball example, order doesn't matter. Instead, we use combinations. When you see permutation problems:

• Consider all the positions each object can take.
• Remember, \(nPr\) is the notation used, where \(n\) is the total number of items and \(r\) is the number of items to arrange.
• The formula is \(nPr = \frac{n!}{(n-r)!}\).

For exercises involving permutations, focus on how the sequence or order of selection affects the result. In the basketball problem, ordering wasn't important, which is why combinations were used instead.

Factorial

Factorial is a fundamental concept in permutations and combinations, often denoted by an exclamation mark (!). It represents the product of all positive integers up to a given number \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow rapidly:

• \( 3! = 6 \)
• \( 4! = 24 \)
• \( 5! = 120 \), and so on.

They're crucial in calculating permutations and combinations. When using combinations, factorial helps calculate how to divide the total arrangements by the orderings within selected groups. Understanding how factorial impacts the calculation will make it clear why we use them in the combination formula, where roles and positions are not a factor.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. It examines how objects can be selected or arranged. In our basketball example, combinatorics helps determine how teams can be formed.
In combinatorics:

• Combinations are used when the sequence doesn't matter.
• You calculate combinations with the formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), representing how many ways you can choose \(k\) items from \(n\).
• Permutations apply when order is significant, though less relevant here, as seen in team selection.

By mastering combinatorics, students can solve problems involving arrangements and selections more effectively, discerning between scenarios where order matters and where it doesn't.

Mathematics Problem Solving

Solving mathematical problems involves understanding concepts and applying them strategically. Begin by analyzing the problem. Determine the key elements and make connections with formulas. For instance, decide whether the problem involves permutations or combinations.
Follow these steps:

• Identify what's being asked. Is order important or not?
• Choose the right method – permutations or combinations?
• Use factorial knowledge to simplify calculations.
• Break down complex problems into simpler parts for easier calculations.

In the basketball team problem, breaking the task into parts—first choosing the center, then the rest—illustrates problem-solving through organization. This structured approach makes it easier to tackle even seemingly complex problems with confidence.