Problem 4 - Algebra for College Students

Question

In how many ways can Arlene, Brent, Carlos, Dave, Ernie, Frank, and Gladys be seated in a row of seven seats so that Arlene and Carlos are side by side? 1440

Step-by-Step Solution

Answer

There are 1440 ways to seat them with Arlene and Carlos together.

Step 1: Understanding the Constraint

We want Arlene and Carlos to be seated next to each other. To handle this constraint, we can treat Arlene and Carlos as a single unit or 'block'. This transforms the problem into arranging 6 blocks rather than 7 individuals.

Step 2: Calculating Block Arrangements

Since Arlene and Carlos form a single block, we need to calculate the number of ways to arrange the 6 blocks (Arlene-Carlos block + 5 other people: Brent, Dave, Ernie, Frank, Gladys). The number of ways to arrange these 6 blocks is given by the factorial of 6, i.e. \(6!\).

Step 3: Calculating Internal Arrangements of the Block

Within their block, Arlene and Carlos can switch places. Therefore, we also need to consider the number of ways to arrange Arlene and Carlos within their block. There are \(2!\) ways to do this since there are 2 people and they can swap places or remain as is.

Step 4: Final Calculation

Multiply the number of arrangements of the blocks by the internal arrangements of the Arlene-Carlos block. This gives the total arrangements as \(6! \times 2! = 720 \times 2 = 1440\) ways.

Key Concepts

Permutations

Permutations refer to the different ways in which a set of objects can be arranged. This is particularly important in problems where the order of arrangement matters, such as seating people or arranging books on a shelf. For example, when arranging 7 people in a line, each different sequence is a unique permutation. The formula for permutations is derived from factorials, which provide the total number of ways to arrange all elements in a set when order is considered. It's crucial to note that permutations consider all possible orders as distinct arrangements.

Factorial

The concept of factorial, represented by an exclamation point (e.g., 6!), is key in determining permutations. It is a mathematical operation where you multiply a series of descending natural numbers. For instance,

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

Factorials are essential when you want to know how many ways you can arrange n items. Whenever you see permutations, factorials are usually not far behind, as they help compute the total number of possible arrangements. Factorials treat each item as distinct, meaning each item's place in an arrangement matters.

Constraints in Arrangements

In many permutation problems, you encounter special constraints. These constraints modify the general rules of permutations and must be addressed to find the correct solution. For example, in this exercise, Arlene and Carlos must be seated next to each other. This constraint turns both individuals into a single unit or 'block.'

• The constraint in the problem reduces the total number of elements to arrange, simplifying calculations.
• You will then arrange all blocks considering each block as a single entity.

Constraints require you to modify the original problem and rethink the typical application of permutations.

Adjacent Positions

Adjacent positions mean that certain elements must be next to each other. In combinatorial problems like this exercise, viewing two adjacent items as a single block helps incorporate constraints into permutation calculations. You treat the block as one and factor in its internal arrangements later.

• In this case, turning Arlene and Carlos into one block changes seven individual elements into six elements (with one being a block).
• Calculate all possible ways to arrange these six elements using permutations.
• Finally, within the single block, consider how Arlene and Carlos might switch places with each other, adding to the overall count of permutations: in this exercise, there are 2 arrangements, as each person can be on either side.

Using blocks is a powerful strategy when handling adjacency in permutations and helps ensure no valid arrangement is overlooked.