How many two-element subsets of {1, 2, 3, 4} are there that do not contain the pair of elements 2...

1. Translate the problem requirements: We need to find how many ways we can choose 2 elements from the set \(\{1, 2, 3, 4\}\), but we cannot choose the specific pair (2, 4) together
2. Count total possible two-element subsets: List all possible ways to choose 2 elements from our 4-element set without any restrictions
3. Identify and exclude the restricted pair: From our complete list, remove any subset that contains both 2 and 4 together
4. Count the remaining valid subsets: Count how many two-element subsets remain after excluding the restricted pair

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to do. We have a set \(\{1, 2, 3, 4\}\) with four numbers in it. We want to find all the ways we can pick exactly 2 numbers from this set, but there's a special rule: we cannot pick both 2 and 4 together in the same subset.

Think of it like choosing 2 people from a group of 4 friends, but there are 2 specific friends who can't be chosen together because they don't get along.

Process Skill: TRANSLATE - Converting the mathematical language of "two-element subsets" and constraints into an everyday scenario we can work with

2. Count total possible two-element subsets

Let's list out every possible way to choose 2 numbers from our set \(\{1, 2, 3, 4\}\). We'll be systematic about this:

Starting with 1: We can pair 1 with each of the remaining numbers

• \(\{1, 2\}\)
• \(\{1, 3\}\)
• \(\{1, 4\}\)

Starting with 2: We can pair 2 with the remaining numbers (but we already counted \(\{1, 2\}\))

• \(\{2, 3\}\)
• \(\{2, 4\}\)

Starting with 3: We can pair 3 with the remaining number (but we already counted pairs with 1 and 2)

• \(\{3, 4\}\)

So our complete list of all possible two-element subsets is:
\(\{1, 2\}\), \(\{1, 3\}\), \(\{1, 4\}\), \(\{2, 3\}\), \(\{2, 4\}\), \(\{3, 4\}\)

That's 6 total subsets.

3. Identify and exclude the restricted pair

Now we need to find which of these subsets violate our rule. The problem says we cannot have "the pair of elements 2 and 4" together.

Looking at our list:

• \(\{1, 2\}\) ✓ (doesn't contain both 2 and 4)
• \(\{1, 3\}\) ✓ (doesn't contain both 2 and 4)
• \(\{1, 4\}\) ✓ (doesn't contain both 2 and 4)
• \(\{2, 3\}\) ✓ (doesn't contain both 2 and 4)
• \(\{2, 4\}\) ✗ (this is exactly the forbidden pair!)
• \(\{3, 4\}\) ✓ (doesn't contain both 2 and 4)

So we need to remove \(\{2, 4\}\) from our list.

Process Skill: APPLY CONSTRAINTS - Carefully checking each subset against the given restriction

4. Count the remaining valid subsets

After removing the forbidden subset \(\{2, 4\}\), we're left with:

• \(\{1, 2\}\)
• \(\{1, 3\}\)
• \(\{1, 4\}\)
• \(\{2, 3\}\)
• \(\{3, 4\}\)

Counting these up: 1, 2, 3, 4, 5 valid subsets.

Final Answer

There are 5 two-element subsets of \(\{1, 2, 3, 4\}\) that do not contain the pair of elements 2 and 4.

The answer is D. Five.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "do not contain the pair 2 and 4"

Students often misread this constraint as "do not contain 2 OR do not contain 4" instead of "do not contain BOTH 2 AND 4 together." This leads them to exclude all subsets containing either element 2 or element 4, which would eliminate \(\{1,2\}\), \(\{1,4\}\), \(\{2,3\}\), \(\{2,4\}\), and \(\{3,4\}\), leaving only \(\{1,3\}\).

2. Forgetting this is a combinations problem

Some students might approach this as a permutations problem, thinking that \(\{2,4\}\) and \(\{4,2\}\) are different subsets. This confusion about whether order matters in subsets can lead to overcounting the total possibilities and getting confused about which restrictions apply.

Errors while executing the approach

1. Missing subsets when listing systematically

When manually listing all possible two-element subsets, students commonly miss one or more combinations. For example, they might forget \(\{3,4\}\) or accidentally skip \(\{1,4\}\), leading to an incorrect total count before even applying the restriction.

2. Incorrectly applying the restriction

Even when students understand the constraint correctly, they might make errors when checking each subset against the rule. They could mistakenly mark \(\{2,4\}\) as valid or incorrectly eliminate a valid subset like \(\{1,4\}\) because it contains the element 4.

Errors while selecting the answer

1. Selecting the count of forbidden subsets instead of valid ones

After correctly identifying that only \(\{2,4\}\) violates the constraint, some students might select "A. One" because there is one forbidden subset, rather than counting the five remaining valid subsets for the final answer.