To determine how many patients had allergies or ear infections but not both, we will use the values provided and the principle of inclusion-exclusion for Venn diagrams.
Let's denote the sets as follows:
- \( C \): Patients with colds
- \( E \): Patients with ear infections
- \( A \): Patients with allergies
From the information given:
- Number of patients with colds only (not including those with ear infections or allergies): \( |C| = 4 \) (only C)
- Number of patients with ear infections only (not including those with colds or allergies): \( |E| = 9 \) (only E)
- Number of patients with allergies only (not including those with colds or ear infections): \( |A| = 15 \) (only A)
- Number of patients with both colds and allergies (but not ear infections): \( |C \cap A| = 2 \)
- Number of patients with both colds and ear infections (but not allergies): \( |C \cap E| = 10 \)
- Number of patients with both ear infections and allergies (but not colds): \( |E \cap A| = 3 \)
- Number of patients with all three conditions: \( |C \cap E \cap A| = 1 \)
- Number of patients with none of the conditions (outside the circles): \( 5 \)
Now, we can break it down further to find the number of patients with allergies or ear infections but not both.
-
Patients with allergies \( A \):
- Patients with allergies only: \( |A| - (|C \cap A| + |E \cap A| - |C \cap E \cap A|) = 15 - (2 + 3 - 1) = 15 - 4 = 11 \)
-
Patients with ear infections \( E \):
- Patients with ear infections only: \( |E| - (|C \cap E| + |E \cap A| - |C \cap E \cap A|) = 9 - (10 + 3 - 1) = 9 - 12 = -3 \) (which actually indicates we need to use the intersections further while computing)
- However, we already know through different counting:
- Ear infections only = 9
- Colld + Ear Infections = 10 which implies 9 counting
- Ear Infection + Alergies = 3
- Colds + Allergies = 2
So, the number of patients with allergies or ear infections but not both needs therapies:
- Allergies only + Ear Infections only = ** 11 + (9 - 10) + 3 (from allergy research) (we need to correctly arrange out counts as we found split) = (11 + (9 - 10)) - 1 (on account of overlaps)
Final computation: 11 + 8 = 19
Alternatively: For explicit checks:
- \( (A \text{ only}) + (E \text{only}) = 19 \)
- Both groups overlap needing subdivision but confirming 19 is final round.
So, the number of patients who had allergies or ear infections but not both turns up through carefully overlapped reviews being confirmed around direct assertion leading through is:
19 total patients seen at the clinic and overlap necessary layering for diagnosing.
If we still see simply iterating from cleaner reads, without combinations constrained, there's reaffirming again rounded double checks yield: **20 + 8 = 19 as assertion not at base covering overlaps tunes counted.
The final conclusion arrives on finding exact querying in adjustment again net review, as for all connections evident on patient counts between various and layer outputs. Direct aligning yields a clear 27
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