A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert...

1. Translate the problem requirements: The restaurant has 6 cheeses and 2 fruits. Each platter must have equal numbers of cheese types and fruit types. We need to find how many different platters are possible.
2. Determine the constraint on platter composition: Since there are only 2 fruits available and each platter needs equal numbers of cheese and fruit types, each platter can have at most 2 types of each.
3. Enumerate the possible platter configurations: Consider each valid combination based on the equal numbers constraint (1 cheese + 1 fruit, or 2 cheeses + 2 fruits).
4. Calculate combinations for each configuration: Use the multiplication principle to count the ways to select the required number of cheese types and fruit types for each valid configuration.
5. Sum all possibilities: Add up the number of different platters from all valid configurations to get the total.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the restaurant is offering and what we need to find.

The restaurant has:

• 6 different kinds of cheese
• 2 different kinds of fruit

Each dessert platter must contain:

• An equal number of cheese types and fruit types
• At least one of each (since it's a dessert platter)

We need to find: How many different dessert platters are possible?

Process Skill: TRANSLATE - Converting the problem language into clear mathematical requirements

2. Determine the constraint on platter composition

Since we only have 2 kinds of fruit available, and each platter needs equal numbers of cheese types and fruit types, let's think about what's actually possible.

If a platter has 1 fruit type, it must have 1 cheese type.
If a platter has 2 fruit types, it must have 2 cheese types.
Can a platter have 3 fruit types? No, because we only have 2 fruits total.

So the only possible platter configurations are:

• Configuration 1: 1 cheese + 1 fruit
• Configuration 2: 2 cheeses + 2 fruits

Process Skill: APPLY CONSTRAINTS - The limited number of fruits restricts our possible combinations

3. Enumerate the possible platter configurations

Now let's think through each valid configuration:

Configuration 1: 1 cheese + 1 fruit
We need to pick 1 cheese from 6 available cheeses, and 1 fruit from 2 available fruits.

Configuration 2: 2 cheeses + 2 fruits
We need to pick 2 cheeses from 6 available cheeses, and 2 fruits from 2 available fruits.

4. Calculate combinations for each configuration

Let's count the possibilities for each configuration using logical reasoning:

Configuration 1: 1 cheese + 1 fruit

• Ways to choose 1 cheese from 6: We can pick any of the 6 cheeses, so that's 6 ways
• Ways to choose 1 fruit from 2: We can pick either of the 2 fruits, so that's 2 ways
• Total platters for this configuration: \(6 \times 2 = 12\) different platters

Configuration 2: 2 cheeses + 2 fruits

• Ways to choose 2 fruits from 2: There's only one way - we must take both fruits
• Ways to choose 2 cheeses from 6: We need to count how many ways we can pick 2 cheeses from 6 available cheeses

For choosing 2 cheeses from 6, let's think systematically:

• If we pick cheese #1, we can pair it with any of the remaining 5 cheeses (5 pairs)
• If we pick cheese #2 (and not #1), we can pair it with any of cheeses #3, #4, #5, #6 (4 pairs)
• If we pick cheese #3 (and not #1 or #2), we can pair it with any of cheeses #4, #5, #6 (3 pairs)
• If we pick cheese #4 (and not #1, #2, or #3), we can pair it with cheese #5 or #6 (2 pairs)
• If we pick cheese #5 (and not #1-#4), we can pair it with cheese #6 (1 pair)

Total ways to choose 2 cheeses: \(5 + 4 + 3 + 2 + 1 = 15\) ways

• Total platters for this configuration: \(15 \times 1 = 15\) different platters

5. Sum all possibilities

Now we add up all the different platters from both valid configurations:

• Platters with 1 cheese + 1 fruit: 12
• Platters with 2 cheeses + 2 fruits: 15
• Total different dessert platters: \(12 + 15 = 27\)

Final Answer

The restaurant could offer 27 different dessert platters.

This matches answer choice E. 27.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint "equal number of kinds"

Students may misunderstand what "equal number of kinds of cheese and kinds of fruit" means. They might think it means equal quantities or weights rather than equal counts of different types. This leads to approaching the problem as a simple multiplication (\(6 \times 2 = 12\)) instead of recognizing it as a combinations problem with constraints.

2. Missing the critical constraint imposed by limited fruit options

Many students fail to recognize that having only 2 kinds of fruit severely limits the possible platter configurations. They might try to consider cases with 3, 4, 5, or 6 types of each item without realizing that you cannot have more fruit types than what's available. This constraint is crucial for determining that only two configurations are possible: (1 cheese + 1 fruit) and (2 cheeses + 2 fruits).

3. Treating this as a permutation problem instead of combinations

Students may incorrectly think that the order in which items are arranged on the platter matters, leading them to use permutation formulas instead of combination formulas. This would result in significantly inflated numbers and wrong answer choices.

Errors while executing the approach

1. Arithmetic errors when calculating combinations manually

When calculating \(\mathrm{C}(6,2) = 15\) by hand using the systematic counting method (\(5+4+3+2+1\)), students often make addition errors or miss some pairs, leading to incorrect totals like 12 or 18 instead of 15.

2. Forgetting to multiply cheese and fruit combinations

For Configuration 1, students may correctly identify 6 ways to choose cheese and 2 ways to choose fruit but forget to multiply them together, reporting 8 (\(6+2\)) instead of 12 (\(6 \times 2\)). This reflects confusion about whether to add or multiply when combining independent choices.

3. Double-counting or missing cases in systematic enumeration

When listing out the ways to choose 2 cheeses from 6, students may accidentally count the same pair twice (like counting both "cheese 1 with cheese 3" and "cheese 3 with cheese 1") or miss some combinations entirely, leading to incorrect subtotals.

Errors while selecting the answer

1. Stopping after calculating only one configuration

Students may correctly calculate either the 12 platters from Configuration 1 or the 15 platters from Configuration 2, but forget that both configurations are valid and need to be added together. This leads them to select answer choice B (12) or C (15) instead of the correct answer E (27).

2. Adding incorrectly when combining final totals

Even after correctly calculating 12 platters for Configuration 1 and 15 platters for Configuration 2, some students make a simple arithmetic error when adding \(12 + 15\), potentially getting 25, 26, or 28, leading them to select the closest available answer choice rather than the correct one.