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A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: Blue, Green, Yellow Or Pink. The store packs the notepads in pacakages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?
Let's break down what we're looking for in everyday language. We have an office supply store that sells notepads, and we need to count how many different types of packages they can create.
The store has:
• 2 different sizes of notepads
• 4 different colors: Blue, Green, Yellow, Pink
Each package contains exactly 3 notepads, and there are only two ways to pack them:
• Type 1: All 3 notepads are the same size AND the same color
• Type 2: All 3 notepads are the same size BUT all 3 different colors
The key constraint is that the order of colors doesn't matter - so a package with Blue-Green-Yellow is the same as Yellow-Blue-Green.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical categories
Now let's think about this systematically. We have two completely separate types of packages to count:
Type 1 packages: Same size, same color
Think of this as: "Give me 3 small blue notepads" or "Give me 3 large pink notepads"
Type 2 packages: Same size, three different colors
Think of this as: "Give me 3 small notepads, each a different color" or "Give me 3 large notepads, each a different color"
Since these are completely different types, we'll count each type separately and then add them together.
For Type 1 packages (same size, same color), let's think step by step:
For each size, we can pick any of the 4 colors. Since all 3 notepads in the package are identical, there's only one way to make each combination.
Small size packages:
• 3 small blue notepads → 1 package type
• 3 small green notepads → 1 package type
• 3 small yellow notepads → 1 package type
• 3 small pink notepads → 1 package type
Total for small size: 4 package types
Large size packages:
• 3 large blue notepads → 1 package type
• 3 large green notepads → 1 package type
• 3 large yellow notepads → 1 package type
• 3 large pink notepads → 1 package type
Total for large size: 4 package types
Total Type 1 packages: \(4 + 4 = 8\)
For Type 2 packages (same size, three different colors), we need to select 3 different colors from our 4 available colors.
Let's list out all the ways to pick 3 different colors from Blue, Green, Yellow, Pink:
• Blue, Green, Yellow
• Blue, Green, Pink
• Blue, Yellow, Pink
• Green, Yellow, Pink
That's 4 different color combinations.
Small size packages:
For each of the 4 color combinations above, we can make 1 package of small notepads.
Total for small size: 4 package types
Large size packages:
For each of the 4 color combinations above, we can make 1 package of large notepads.
Total for large size: 4 package types
Total Type 2 packages: \(4 + 4 = 8\)
Process Skill: CONSIDER ALL CASES - Systematically listing all possible color combinations
Now we add up both types of packages:
Type 1 packages (same size, same color): 8
Type 2 packages (same size, different colors): 8
Total different packages possible: \(8 + 8 = 16\)
The answer is 16 different packages, which corresponds to choice C.
Let's verify this makes sense: We found 8 ways to make same-color packages (\(4 \times 2\) sizes) and 8 ways to make different-color packages (\(4 \times 2\) sizes), giving us 16 total package types.
1. Misinterpreting the constraint about different colors in Type 2 packages
Students often misread "3 notepads of the same size and of 3 different colors" as meaning they can pick any 3 colors with repetition allowed, rather than understanding that all 3 colors must be different. This leads them to incorrectly include cases like "Blue, Blue, Green" in their Type 2 count.
2. Overlooking that order doesn't matter
The problem states "the order in which the colors are packed is not considered," but students frequently miss this constraint. They might count "Blue-Green-Yellow" and "Green-Blue-Yellow" as different packages, leading to significant overcounting in Type 2 packages.
3. Confusing the two package types or missing one entirely
Students sometimes fail to recognize that there are exactly two distinct package categories. They might try to create a third type like "same color, different sizes" or completely miss counting one of the two types, focusing only on same-color packages or only on different-color packages.
1. Incorrectly calculating combinations for Type 2 packages
When finding how many ways to choose 3 different colors from 4 available colors, students often make calculation errors. They might incorrectly use permutations instead of combinations (getting 24 instead of 4), or they might miscalculate the combination formula \(\mathrm{C(4,3)} = 4\).
2. Forgetting to account for both sizes in each package type
Students frequently calculate the color combinations correctly but forget that each combination needs to be counted twice - once for small size and once for large size. For example, they might find 4 Type 1 packages and 4 Type 2 packages, forgetting to double each for the two sizes.
3. Double-counting or missing cases when listing systematically
When manually listing out the possible color combinations, students often either skip combinations (like missing "Green, Yellow, Pink") or accidentally count the same combination twice due to different orderings.
1. Adding the subtotals incorrectly
Even after correctly calculating 8 Type 1 packages and 8 Type 2 packages, students sometimes make simple arithmetic errors when adding \(8 + 8\), or they might add intermediate values incorrectly (like adding \(4 + 4 + 4\) instead of recognizing the final total should be 16).