Problem 40 - College Algebra with Corequisite Support

Question

A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back. (a) How many arrangements are possible with no restrictions? (b) How many arrangements are possible if the parents must sit in the front? (c) How many arrangements are possible if the parents must be next to each other?

Step-by-Step Solution

Answer

(a) 120 arrangements; (b) 6 arrangements; (c) 48 arrangements.

Step 1: Total Arrangements

First, calculate the total number of ways to arrange 2 members in the front and 3 members in the back from 5 family members. This can be done by selecting 2 members for the front out of 5 and placing the remaining 3 in the back. Use the combination formula \( \binom{n}{r} \):Calculate \( \binom{5}{2} = 10 \), which represents the ways to choose who goes in the front.Each group (front and back) can be arranged internally:- Arrange 2 people in the front: there are \( 2! = 2 \) ways.- Arrange 3 people in the back: there are \( 3! = 6 \) ways.Thus, total arrangements = \( 10 \times 2! \times 3! = 120 \).

Step 2: Arrangements with Parents in Front

If the parents must be in the front, their positions are fixed. Calculate the arrangements for the children in the back:- The 3 children can be arranged in \( 3! \) different ways, because all must sit in the back.Therefore, the total arrangements with parents in the front: \( 3! = 6 \).

Step 3: Arrangements with Parents Next to Each Other

Consider the parents as a single unit or block in the arrangement. Now we arrange 4 units (2 parents as one unit and 3 children). The steps are:- There are \( 4! \) ways to arrange 4 units.- Within the parents' block, the parents can be arranged in \( 2! \) ways.Thus, total arrangements with parents next to each other = \( 4! \times 2! = 48 \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arranging, and combination of objects. It helps us understand how many ways we can organize a distinct set of items or people according to specific rules. In the context of the family photo exercise, combinatorics helps us determine the different ways to organize family members for their picture.

For example, when calculating the number of ways to choose 2 family members to stand in the front, we use combinatorics to select any 2 from the 5 members. The combination formula \( \binom{n}{r} \) is utilized, where \( n \) is the total number of items, and \( r \) is the number selected. So, \( \binom{5}{2} \) helps find how many ways you can pick 2 people for the front row, which comes out to 10.

Once chosen, combinatorics also helps organize these selected members internally, by arranging them in different orders within their specific places, both at the front and back.

Factorials

Factorials are a crucial mathematical function often used in permutations and combinations. They simplify the process of finding possible arrangements of a set of items. The factorial of a number \( n \), represented as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 3! = 3 \times 2 \times 1 = 6 \).

In the family picture problem, factorials help calculate the arrangements within groups. After selecting 2 family members for the front, we use \( 2! = 2 \) to find out in how many ways these two members can be organized. Similarly, \( 3! = 6 \) is used to determine how the 3 members at the back can be arranged.

• This sequential approach simplifies several calculations related to ordering people or items in a meaningful sequence.
• Factorials are common when you deal with permutations with no repeated items as each object's position is unique.

Naturally, they are a foundational tool in combinatorics, streamlining organization possibilities.

Family Arrangements

Arranging family members for a photo can involve various rules or conditions, which leads to different arrangement calculations. In this exercise, the number of arrangements changes based on whether certain members need to be placed at specific spots or if a group should remain together.

For instance, if no restrictions apply, our main focus is purely on the number of ways to choose and arrange these 5 members at specified spots, resulting in a large number of configurations.

When we specify that parents should be at the front, it restricts their positions to just the two front spots, drastically reducing the arrangements that are possible for the rest of the members.

Alternatively, when considering parents as a block of 2, we only have 4 blocks to arrange - the parent block and 3 individual children. This condition changes the count using a lower number of factorial calculations:\( 4! \) for arranging blocks and then \( 2! \) for arranging within the parent block. In this manner, different "rules" or "conditions" create distinct arrangement methods, showcasing flexibility in pattern and order creation.