Diana designs stained-glass windows. She has a large supply of square glass window panes. Some panes are yellow, and some panes are red. She wants to place them in windows that are 3x2 in such a way that no two windows are the same, even when the design is rotated or reflected. How many different window designs can she create using any combination of red and yellow panes?
To determine the number of different window designs Diana can create using the red and yellow panes, we can break down the problem step-by-step:
Step 1: Determine the number of yellow panes Diana can use.
Since she has a large supply of square glass window panes, the number of yellow panes she can use can vary. For this step-by-step analysis, let's assume she has a total of n yellow panes.
Step 2: Determine the number of red panes Diana can use.
Similarly, the number of red panes Diana can use is variable. For this analysis, let's assume she has a total of m red panes.
Step 3: Calculate the total number of possible window designs.
To do so, we need to determine the number of ways we can arrange m red panes and n yellow panes in a 3x2 window.
To create a 3x2 window design, there are two possible scenarios:
- The window design consists of three horizontal panes (3 yellow or 3 red).
- The window design consists of two vertical panes (2 yellow, 1 red OR 2 red,1 yellow).
Let's calculate the number of possibilities for each scenario:
Scenario 1: Three Horizontal Panes (3 yellow or 3 red)
Since the window design consists of three horizontal panes, there are only two options: 3 yellow or 3 red. So, this scenario has 2 possibilities.
Scenario 2: Two Vertical Panes (2 yellow, 1 red OR 2 red, 1 yellow)
To calculate the number of possibilities for this scenario, we need to consider the number of ways we can arrange 2 yellow panes and 1 red pane, as well as the number of ways to arrange 2 red panes and 1 yellow pane.
- Number of ways to arrange 2 yellow panes and 1 red pane:
This can be calculated using combinations. We need to select 2 panes out of n (yellow) and 1 pane out of m (red). The formula for combinations is nCr, where n is the total number of items and r is the number of items to be selected. For this scenario, the number of possibilities is given by: nC2 * mC1.
- Number of ways to arrange 2 red panes and 1 yellow pane:
Similarly, the number of possibilities for this scenario is given by: mC2 * nC1.
Since these two vertical pane scenarios are distinct (one has 2 yellow and 1 red, and the other has 2 red and 1 yellow), we need to add them together to calculate the total number of possibilities for this scenario: nC2 * mC1 + mC2 * nC1.
Step 4: Calculate the total number of window designs.
To determine the total number of possible window designs, we need to sum up the possibilities from both scenarios:
Total = Scenario 1 possibilities + Scenario 2 possibilities
Total = 2 + (nC2 * mC1 + mC2 * nC1).
Substituting n and m with the actual numbers of yellow and red panes, you can calculate the total number of different window designs Diana can create using any combination of red and yellow panes.
To determine the number of different window designs Diana can create using red and yellow panes, we need to consider the arrangements of the panes.
Let's break down the problem into smaller steps:
Step 1: Count the number of possibilities for each individual row.
Since each row has two panes, there are 2 possibilities: either both panes are red or one pane is red and the other is yellow.
Step 2: Determine the number of distinct combinations of rows.
Since there are 3 rows, we need to consider the possibilities for each row and multiply them together. For example, if there are 2 possibilities for each row, there would be 2 * 2 * 2 = 8 distinct combinations.
However, we have to take into consideration that we want each window design to be unique even when rotated or reflected. To do this, we need to divide the number of distinct combinations by the number of possible rotations and reflections.
Step 3: Count the number of possible rotations and reflections.
Let's consider a single window with rows labeled as A, B, and C:
A B
C D
There are four possible rotations:
1. Original:
A B
C D
2. 90 degrees clockwise rotation:
C A
D B
3. 180 degrees clockwise rotation:
D C
B A
4. 270 degrees clockwise rotation:
B D
A C
In addition to the rotations, there are two possible reflections:
1. Reflection along the vertical axis:
B A
D C
2. Reflection along the horizontal axis:
C D
A B
Therefore, there are a total of 4 rotations plus 2 reflections, which gives us a total of 6 possible rotations and reflections.
Step 4: Calculate the final number of different window designs.
To obtain the final number of different window designs, divide the number of distinct combinations of rows by the number of possible rotations and reflections.
So, the final answer is:
Number of different window designs = (Number of possibilities for each row)^(Number of rows) / (Number of possible rotations and reflections)
In this case, since each row has 2 possibilities (red or yellow), and there are 3 rows, the calculation would be:
Number of different window designs = (2^3) / 6 = 8 / 6 = 1.33
Since we can't have a fraction of a window design, we have to round down to the nearest whole number. Therefore, Diana can create a total of 1 different window design using any combination of red and yellow panes.