- Translate the problem requirements: We need to find the minimum possible number of applicants who were recommended by all three interviewers, given that 15 were recommended by interviewer 1, 17 by interviewer 2, and 20 by interviewer 3, out of 30 total applicants.
- Identify the constraint boundaries: Determine how many applicants were NOT recommended by each interviewer to understand the limits of non-overlap.
- Construct the minimum overlap scenario: Arrange the recommendations to minimize intersection by maximizing the number of applicants who receive recommendations from only one or two interviewers.
- Verify feasibility: Check that our minimum scenario doesn't violate the given constraints about total recommendations per interviewer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what we're looking for. We have 30 applicants total, and three interviewers who each make recommendations:
- Interviewer 1 recommended 15 applicants
- Interviewer 2 recommended 17 applicants
- Interviewer 3 recommended 20 applicants
The question asks: What's the minimum number of applicants who could have been recommended by all three interviewers?
Think of it this way: we want to arrange these recommendations so that as few people as possible get recommended by all three. This means we want to spread out the recommendations as much as possible.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding of what we're minimizing
2. Identify the constraint boundaries
To minimize overlap, let's think about who is NOT recommended by each interviewer:
- Interviewer 1 did NOT recommend: \(30 - 15 = 15\) applicants
- Interviewer 2 did NOT recommend: \(30 - 17 = 13\) applicants
- Interviewer 3 did NOT recommend: \(30 - 20 = 10\) applicants
Here's the key insight: If someone is NOT recommended by any interviewer, they definitely cannot be in the group recommended by all three. So the more people we can place in the "not recommended" categories, the fewer will be left to potentially overlap.
3. Construct the minimum overlap scenario
To minimize the intersection of all three sets, let's try to maximize how many people are excluded from the "all three" group.
The best case scenario would be if we could arrange things so that anyone who gets excluded by at least one interviewer covers as many people as possible.
Let's see if we can make it so that NO ONE gets recommended by all three:
- 15 people are not recommended by interviewer 1
- 13 people are not recommended by interviewer 2
- 10 people are not recommended by interviewer 3
If these groups of "not recommended" people are completely different (no overlap between them), then we would exclude \(15 + 13 + 10 = 38\) people from being recommended by all three.
But wait! We only have 30 people total. Since \(38 > 30\), this means these "not recommended" groups must overlap.
However, even with some overlap, we can still potentially exclude everyone. If the \(15 + 13 + 10 = 38\) "exclusions" can cover all 30 people (with some people being excluded by multiple interviewers), then it's possible that zero people are recommended by all three.
Process Skill: CONSIDER ALL CASES - Exploring the extreme case to find the absolute minimum
4. Verify feasibility
Let's check if having zero people recommended by all three is actually possible:
If 0 people are recommended by all three, then every person is missing at least one recommendation. We need:
- 15 people missing interviewer 1's recommendation
- 13 people missing interviewer 2's recommendation
- 10 people missing interviewer 3's recommendation
- Total "missing recommendations" = \(15 + 13 + 10 = 38\)
Since we have 30 people, and we need 38 "missing recommendation slots," some people must be missing recommendations from multiple interviewers. This is perfectly fine and doesn't create any contradictions.
For example, one valid arrangement could be:
- 8 people get no recommendations at all (missing all three)
- 7 people get only interviewer 1's recommendation (missing interviewers 2 and 3)
- 5 people get only interviewer 2's recommendation (missing interviewers 1 and 3)
- 10 people get only interviewer 3's recommendation (missing interviewers 1 and 2)
Let's verify: \(8 + 7 + 5 + 10 = 30\) people ✓
Interviewer 1 recommends: 7 + 0 + 0 + 0 = 7... wait, this doesn't work.
Let me reconsider with a working arrangement:
- 5 people recommended by interviewers 1 and 2 only
- 10 people recommended by interviewers 1 and 3 only
- 12 people recommended by interviewers 2 and 3 only
- 3 people recommended by interviewer 1 only
- 0 people recommended by interviewer 2 only
- 0 people recommended by interviewer 3 only
- 0 people recommended by all three
- 0 people with no recommendations
This gives us \(5 + 10 + 12 + 3 + 0 + 0 + 0 + 0 = 30\) people ✓
Recommendations by each interviewer:
- Interviewer 1: \(5 + 10 + 3 = 18\)... this is too many (should be 15)
Actually, let me use a simpler approach: since the total "missing recommendations" (38) exceeds the number of people (30), it's mathematically possible for everyone to be missing at least one recommendation. Therefore, 0 people recommended by all three is achievable.
5. Final Answer
The minimum number of applicants who could have been recommended by all three interviewers is 0.
This corresponds to answer choice A. 0.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the optimization direction: Students often confuse "minimum overlap" with "maximum overlap" and try to maximize the number of people recommended by all three interviewers instead of minimizing it. This leads them to look for the maximum possible intersection rather than exploring whether zero intersection is possible.
- Incorrectly applying the inclusion-exclusion principle: Students may immediately jump to using the formula \(|A \cap B \cap C| = |A| + |B| + |C| - |A \cup B \cup C| + \text{other terms}\), without recognizing that this gives them a specific value rather than helping them find the minimum possible value through strategic arrangement of the sets.
- Failing to consider the constraint flexibility: Students may assume that the given numbers (15, 17, 20 recommendations) create rigid constraints that force a certain minimum overlap, without realizing that these totals can be achieved through many different arrangements of individual recommendations.
Errors while executing the approach
- Arithmetic errors in verification: When students attempt to construct a specific arrangement to verify that zero overlap is possible (as shown in the solution), they often make calculation errors while checking that their proposed distribution actually yields the required recommendation totals (15, 17, 20) for each interviewer.
- Incomplete feasibility checking: Students may conclude that zero overlap is impossible after trying one or two arrangements that don't work, without systematically exploring whether alternative arrangements could achieve the desired result. They give up too early in the verification process.
- Mishandling the "missing recommendations" logic: When using the approach of counting people NOT recommended by each interviewer (15, 13, 10), students may incorrectly conclude that since \(15 + 13 + 10 = 38 > 30\), there must be some minimum overlap, without recognizing that this actually supports the possibility of zero intersection.
Errors while selecting the answer
- Defaulting to a "reasonable" non-zero answer: Even after correctly determining that zero overlap is mathematically possible, students may doubt this result and choose a small positive number like 2 or 3, thinking that zero seems "too good to be true" or unrealistic in a practical scenario.
