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How many two-element subsets of \(\{1, 2, 3, 4\}\) are there that do not contain the pair of elements 2 and 4?
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
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
Starting with 2: We can pair 2 with the remaining numbers (but we already counted \(\{1, 2\}\))
Starting with 3: We can pair 3 with the remaining number (but we already counted pairs with 1 and 2)
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.
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:
So we need to remove \(\{2, 4\}\) from our list.
Process Skill: APPLY CONSTRAINTS - Carefully checking each subset against the given restriction
After removing the forbidden subset \(\{2, 4\}\), we're left with:
Counting these up: 1, 2, 3, 4, 5 valid subsets.
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.
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 \(\{1,3\}\).
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.
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.
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.
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.