Questions: This exercise can be solved using combinations even though the problem statement will not always include a form of the word "combination" or "subset." Kierman (the contractor) is to build eight homes on a block in a new subdivision, using two different models: standard and deluxe. (All standard homes are the same, and all deluxe models are the same.) (a) How many different choices does Kiernan have in positioning the eight houses if they decide to build five standard and three deluxe models? (b) If Kiernan builds four deluxes and four standards, how many different positionings can they use?

This exercise can be solved using combinations even though the problem statement will not always include a form of the word "combination" or "subset."

Kierman (the contractor) is to build eight homes on a block in a new subdivision, using two different models: standard and deluxe. (All standard homes are the same, and all deluxe models are the same.)
(a) How many different choices does Kiernan have in positioning the eight houses if they decide to build five standard and three deluxe models?
(b) If Kiernan builds four deluxes and four standards, how many different positionings can they use?

Transcript text: This exercise can be solved using combinations even though the problem statement will not always include a form of the word "combination" or "subset." Kierman (the contractor) is to build eight homes on a block in a new subdivision, using two different models: standard and deluxe. (All standard homes are the same, and all deluxe models are the same.) (a) How many different choices does Kiernan have in positioning the eight houses if they decide to build five standard and three deluxe models? (b) If Kiernan builds four deluxes and four standards, how many different positionings can they use?

Solution

Solution Steps

Solution Approach

To solve these problems, we need to determine the number of ways to arrange a certain number of standard and deluxe homes. This is a classic combinations problem where we choose positions for one type of home, and the remaining positions will automatically be for the other type. We can use the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of positions and \( k \) is the number of positions to choose for one type of home.

(a) For five standard and three deluxe models, we choose 5 positions out of 8 for the standard homes.

(b) For four deluxe and four standard models, we choose 4 positions out of 8 for the deluxe homes.

Step 1: Determine Choices for Standard and Deluxe Models

To find the number of different choices Kiernan has in positioning the eight houses with five standard and three deluxe models, we use the combination formula:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

where \( n = 8 \) (total houses) and \( k = 5 \) (standard houses). Thus, we calculate:

\[ C(8, 5) = \frac{8!}{5! \cdot (8-5)!} = \frac{8!}{5! \cdot 3!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56 \]

Step 2: Determine Choices for Equal Models

Next, we calculate the number of different choices when Kiernan builds four deluxe and four standard models. Again, we apply the combination formula:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Here, \( n = 8 \) and \( k = 4 \) (deluxe houses). Thus, we compute:

\[ C(8, 4) = \frac{8!}{4! \cdot (8-4)!} = \frac{8!}{4! \cdot 4!} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1} = 70 \]