Problem 67 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Arranging a Class Picture In how many ways can ten students be arranged in a row for a class picture if John and Jane want to stand next to each other, and Mike and Molly also insist on standing next to each other?

Step-by-Step Solution

Answer

There are 161280 ways to arrange the students.

Step 1: Treating Specific Pairs as Single Entities

Since John and Jane want to stand next to each other, treat them as a single unit, or 'block'. Similarly, treat Mike and Molly as another block. This reduces the problem to arranging these two blocks within the larger set of students.

Step 2: Counting the Blocks

Since there are 10 students, treating John-Jane as one block and Mike-Molly as another block leaves us with 10 - 2 pairs + 2 'blocks' = 8 entities to arrange: 6 individuals + 2 blocks.

Step 3: Arranging the Units

Calculate the number of ways to arrange these 8 units. The number of permutations of 8 distinct items is given by the factorial of 8: \(8!\).

Step 4: Arranging Individuals Within Blocks

Within the John-Jane block, John and Jane can switch places, giving us 2 additional arrangements. Likewise, within the Mike-Molly block, there are 2 ways to arrange them as well. Therefore, multiply by 2 for each block.

Step 5: Calculating the Total Arrangements

Multiply the total permutations of the blocks by the internal permutations of each block: \(8! \times 2 \times 2\). Thus, the total number of arrangements is \(8! \times 4\).

Step 6: Calculating 8! and Final Answer

Calculate \(8! = 40320\). Therefore, the total number of arrangements is \(40320 \times 4 = 161280\).

Key Concepts

Factorials

Factorials are a fundamental concept in permutations and counting principles. A factorial is the product of all positive integers from 1 to a given number. It is denoted by the symbol

• "!" after a number. For example, the factorial of 5 is written as 5! and calculated as follows:5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.

Factorials are extremely useful in determining how many ways distinct objects can be arranged in a sequence. In general, the factorial of a number \(n\) gives you the number of permutations of \(n\) distinct objects:

• \(n! = n \cdot (n-1) \cdot (n-2) \cdots 1\).

In our exercise, we calculated the factorial of 8 (or 8!), which stands for the total number of ways to arrange 8 units, derived from treating certain pairs as single blocks. Understanding factorials simplifies complex permutation problems considerably.

Arrangement of Objects

When we talk about the arrangement of objects, we are referring to permutations. This involves selecting and organizing objects from a particular set in a specific order. However, problems become more interesting when certain conditions are imposed, like specific objects needing to be next to one another. This means we treat these specific groups as single units.

• In our example, with John and Jane wanting to be together, and Mike and Molly requiring the same, we treated them as individual blocks.

This transforms the problem into arranging fewer "blocks" or "entities" rather than ten individual students, which makes calculations more manageable. By recognizing these pairings as singular units, we're reorganizing existing objects into a simpler set while respecting the conditions provided.

Counting Principles

Counting principles are key techniques used to solve problems involving the organization and arrangement of items. These principles help us systematically count the number of possible arrangements or combinations. To properly use counting principles:

• Identify distinct groups or units that can be treated singularly for ease of arrangement.
• Calculate arrangements of these units — using permutations where order matters.

The counting principle was applied in our example by ensuring the use of factorials for orderly arrangement, and by multiplying internal permutations within the blocks. For instance, with John and Jane, and Mike and Molly as blocks, we used the multiplication principle: arrange the larger blocks first and then multiply by the arrangements within each block. The principle helps demystify complex arrangements by breaking the problem into smaller, manageable parts that respect initial conditions and any imposed constraints.