Question
Three different colors of paint are available to paint the window frame, the door, and the wall of a house. If no two parts of the house may be painted the same color, how many different ways are there to paint the three parts of the house?
- A3
- B6
- C9
- D27
Show the answer
Correct answer: B
One method of solving the problem is to use a table to represent the sample space. Let the three colors be red, blue, and green.
PAINT COLORS
| Window Frame | Door | Wall |
|---|---|---|
| Red | Blue | Green |
| Red | Green | Blue |
| Blue | Red | Green |
| Blue | Green | Red |
| Green | Red | Blue |
| Green | Blue | Red |
There are 6 possible combinations of colors for the window frame, the door, and the wall where no two parts are painted the same color.
Question analysis
Three paint colors are available for a house's window frame, door and wall, and no two of the three parts may be painted the same color. Students count the different ways to paint them. It tests counting arrangements with a sample space, and the rule against repeating a color is what shapes the count.
- NYS standards
- 7.SP.8b sample spaces for compound events
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Counting arrangements with an organized list is grade 7 work, and this one has only three parts and three colors. It is typical practice, with the no-repeat rule as the one condition to keep in mind.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 8, 47 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
With three colors and three parts, the six arrangements can be listed in well under a minute, so the question rewards either a quick organized list or the counting principle. The main loss comes from ignoring the rule that no two parts match, which turns 3 × 2 × 1 into 3 × 3 × 3. The other wrong choices simply combine the two 3s in the stem.
Something you must already know minor Working it out
The counting principle is assumed: 3 color choices for the window frame, 2 left for the door, then 1 for the wall, multiplied together. (Anderson & Krathwohl, 2001)
Turning words into math minor Understanding the question
The rule that "no two parts of the house may be painted the same color" means each color is used exactly once, which removes one choice at every step. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Strategy: Count the color choices for the window frame, then for the door with one color already used, then for the wall, and multiply; or list the arrangements.
Traps
- Without replacement: In a second pick there is one fewer item, which changes the probability.
- Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.
The wrong answers, classified
A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Gives the number of colors, 3, as the number of ways, without arranging the colors across the three parts.
C Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Multiplies the 3 colors by the 3 parts, 3 × 3 = 9, a product of the stem's numbers rather than a count of arrangements.
D Without replacement (Fischbein & Schnarch, 1997)
Allows any color on each part, 3 × 3 × 3 = 27, ignoring the rule that a color used once cannot be used again.
Worked solution, step by step
Formulas used
- Counting principle
- a is the number of choices for the first part, b for the second and c for the third. Multiply them to get every possible way. It works because each choice for one part can go with each choice for the next part.
Sort out the question
There are 3 colors and 3 parts to paint: the window frame, the door and the wall. No two parts may be the same color. So each color is used exactly once.
You need the number of different ways to paint the three parts.
Count the choices for each part
Paint the parts in order. Each color you use is gone for the next part.
- Window frame: any of the 3 colors, so 3 choices.
- Door: one color is used, so 2 choices are left.
- Wall: only 1 color is left, so 1 choice.
Multiply the choices
Use the counting principle. Each choice for the window frame goes with each choice for the door, and so on. So you multiply.
Watch out: , choice D, lets a color be used twice. , choice C, just multiplies the numbers in the question.
So the answer is B, 6.
Check your answer
List them. Use R for red, B for blue and G for green, in the order frame, door, wall: RBG, RGB, BRG, BGR, GRB and GBR. That is 6 ways, and each one uses all three colors once.
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