Problem 47 - Algebra 2

Question

Indicate whether each situation involves a combination or a permutation. 12 books arranged on a shelf

Step-by-Step Solution

Answer

The situation involves a permutation.

Step 1: Understand the Difference Between Combinations and Permutations

A permutation is an arrangement of objects where the order is important. A combination is a grouping of objects where the order does not matter.

Step 2: Analyze the Situation

Since the books will be arranged on a shelf, the order in which they are arranged matters. Therefore, the situation involves arranging items, and the order of arrangement is important.

Step 3: Determine the Correct Concept

Because the order in which the books are arranged on the shelf matters, this situation involves a permutation, not a combination.

Key Concepts

Permutation Definition

When we talk about permutations, we're discussing the various ways in which a set of objects can be arranged in a sequence where the order of those objects is significant. To illustrate with a simple example, if we have three different books—A, B, and C—a permutation would take into consideration the different ways they can be ordered: ABC, ACB, BAC, BCA, CAB, or CBA are all unique permutations.

In more mathematical terms, a permutation can be calculated using the formula: \(n!\), where 'n' represents the total number of objects, and '!' denotes factorial, meaning you multiply the number by every positive integer less than itself down to one. Suppose we want to find out how many ways we can arrange 5 books, we would use \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) arrangements. This approach, however, assumes that we want to arrange all of the books. If we only want to arrange a subset, the formula adjusts to \( \frac{n!}{(n - r)!} \), where 'r' is the number of objects we wish to select.

Combination Definition

Moving on to combinations, this concept differs from permutations in that the order of the objects does not matter. Suppose you have a salad and you want to include some toppings from a selection of five different items. If you choose three, it does not matter if you select tomatoes, cheese, and olives or cheese, olives, and tomatoes; the combination is the same. The formula used to calculate combinations is: \( \frac{n!}{r! \times (n - r)!} \), where 'n' is the total number to choose from, and 'r' is the number of items being chosen.

For instance, if we wish to choose 3 books to read out of 5 on a shelf, but we don't care in which order we read them, we would calculate this as \( \frac{5!}{3! \times (5 - 3)!} \). Simplified, this gives us \( \frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} = 10 \) possible combinations for which three books we could begin reading.

Arranging Objects in Order

The concept of arranging objects in order comes into play when the position of each item is crucial, which directly ties in with permutations, as shown in the previous example of arranging books on a shelf. Whenever an arrangement has a specific sequence that must be followed or where the position of each element affects the outcome, it is essential to factor in the ordering.

For instance, think about a race with three participants. It matters greatly whether a runner comes in 1st, 2nd, or 3rd. The different possible orders of finishers are distinct outcomes that would be calculated as permutations. Emblematic applications of ordering objects extend to passwords, lock combinations where the sequence matters, and the seating arrangement of guests at a wedding where the order reflects social aspects. Understanding when and why it is necessary to arrange objects in a specific order is paramount for discerning between the need to calculate permutations or combinations.