Problem 27 - Algebra 2

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. choosing two CDs to buy from ten that are on sale

Step-by-Step Solution

Answer

This is a combination with 45 possibilities.

Step 1: Understand the Difference

Permutations and combinations are both ways of counting arrangements or selections but with a key difference: for permutations, the order matters, while for combinations, it does not. In this problem, we are selecting CDs to buy, and the order in which we buy them doesn't matter. Therefore, it's a problem of combinations.

Step 2: Identify Parameters

We need to choose 2 CDs from a total of 10 CDs. This identifies our selection process: we are picking a smaller group (2 CDs) from a larger group of items (10 CDs).

Step 3: Apply the Combination Formula

The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 10 \) and \( r = 2 \).

Step 4: Calculate the Number of Combinations

Substitute the given values into the formula: \[ C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]Thus, there are 45 ways to choose 2 CDs from the 10 on sale.

Key Concepts

Permutations

Permutations are a fundamental concept in combinatorial mathematics where the arrangement of objects matters. Imagine you have a unique code for a lock, and the sequence of numbers must be in the right order to unlock it. That's a permutation. Every different way you can arrange a set of items is a distinct permutation. For example, if you were arranging three books on a shelf, each specific arrangement is a unique permutation, say book A first, B second, and C third is different from B first, A second, and C third.

In mathematical terms, permutations are calculated using the factorial function, denoted by an exclamation point (!). For example, the number of ways to arrange 4 items is given by the factorial 4!, which is 4 × 3 × 2 × 1 = 24 ways.

• Order matters in permutations.
• Calculated using factorials.
• Useful for scenarios like arranging schedules or ordering tasks.

Combinatorial Mathematics

Combinatorial mathematics is the branch of mathematics that deals with counting and arrangement of objects within a set. It has a wide application in fields like cryptography, network theory, and probability. At its core, combinatorial mathematics helps us solve problems related to selecting, arranging, and counting configurations of elements.

In the exercise mentioned, combinatorial mathematics enables us to identify whether a problem involves permutations or combinations. Understanding the fundamentals of this area empowers learners to approach complex selection and representation issues methodically. Here are some key aspects:

• Deals with arrangements where both order and selection can matter.
• Uses mathematical tools such as permutations and combinations to calculate possible arrangements or selections.
• Help solve real-world scheduling, optimization, and probability problems.

Counting Principles

Counting principles form the foundational approach in combinatorial mathematics, aiding in determining the number of possible outcomes in various scenarios. The two fundamental counting principles include the *addition principle* and the *multiplication principle*.

The addition principle is used when there's a choice between events. For example, if you have 3 choices of desserts and 2 choices of drinks, but you can only pick one, you add the number of choices: 3 + 2 = 5 choices.

The multiplication principle is used when events happen in order. If you want to pick one dessert and one drink from the same options, you multiply the number of choices: 3 desserts × 2 drinks = 6 combinations.

• **Addition Principle**: Used when one of several choices is to be selected.
• **Multiplication Principle**: Applied when making multiple selections that occur independently of one another.
• These principles simplify complex combinatorial problems into manageable calculations.