Problem 71 - Algebra and Trigonometry

Question

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

161280 ways.

Step 1: Group John and Jane Together

Since John and Jane must stand next to each other, we can treat them as a single unit or "block." This reduces the problem to finding the number of ways to arrange nine units (eight students + John and Jane as one).

Step 2: Group Mike and Molly Together

Similarly, Mike and Molly must also stand next to each other, so we treat them as another single unit. This further reduces the problem to arranging eight units (seven students + John and Jane block + Mike and Molly block).

Step 3: Calculate Arrangements for Combined Units

Now, calculate the ways to arrange these eight units. The number of ways is given by the factorial of 8, which is calculated as \(8!\).

Step 4: Account for Internal Arrangements of Blocks

Within the John and Jane block, two possible arrangements are (John, Jane) or (Jane, John). Therefore, there are 2 arrangements. Similarly, within the Mike and Molly block, there are also 2 arrangements: (Mike, Molly) or (Molly, Mike).

Step 5: Combine All Possibilities

Now multiply the number of ways to arrange the eight units by the number of internal arrangements for both blocks: \(8! \times 2 \times 2\). This calculation gives the total number of arrangements.

Step 6: Calculate the Final Result

Compute the value: \(8! = 40320\). Therefore, the total number of ways to arrange the students is \(40320 \times 2 \times 2 = 161280\).

Key Concepts

Factorial Calculation

The concept of factorial calculation is a fundamental idea in permutations and combinations. When you see a number followed by an exclamation mark, such as \( n! \), it represents the product of all positive integers from 1 to \( n \). For instance, \( 8! \) means \( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \). Factorials are used to calculate the total number of ways to arrange a set of items. This operation quickly grows in value, emphasizing how many different arrangements can be made as you increase the number of items. For example:

• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

The factorial function is essential when dealing with permutations because it helps in calculating the total possible orders in which a certain number of elements can appear. Understanding how to compute a factorial is crucial when solving problems that involve arranging or ordering objects.

Grouping Principle

The grouping principle is a clever technique in permutations, which simplifies complex problems by considering groups or blocks as single units. When individuals in a set have specific requirements, such as wanting to remain together, treating them as one block can make the calculation more manageable.

• For example, if two students, John and Jane, wish to stand next to each other in a photo, they can be treated as a single unit or block. This concept is similar to creating a cluster of elements.
• In the problem, two such blocks are formed – one for John and Jane, and another for Mike and Molly. This originally reduced the problem from ten individuals to eight units (including the blocks).

By reducing the problem's complexity this way, you effectively change it into arranging a smaller number of entities, where each group can have its own internal arrangement. This simplification not only makes the problem easier to visualize but also lessens the calculation required to find the total number of permutations.

Counting Principles

Counting principles are at the core of finding out the different arrangements or selections in a given problem. The basic idea is to systematically account for all possibilities without redundancy or omission. Here is a structured approach:

• Start by determining the number of elements to arrange or choose from.
• Apply the factorial method to find the permutations of the distinct units when ordering matters.
• When specific groups need to be made, use the grouping principle to understand how many units are being arranged.
• Finally, account for internal arrangements within each block: this requires multiplying the permutations of individual blocks internally. For example, within a block containing John and Jane, two possibilities exist: (John, Jane) and (Jane, John).

In this exercise, the final calculation involves multiplying the factorial value of the total units (eight) by the internal arrangements possible within each group (John/Jane and Mike/Molly), resulting in the total possible arrangements. Understanding and applying these counting principles allow for a consistent and error-free approach to such problems.