Sixteen patients in a hospital must undergo a blood test for a disease. It is known that exactly one of them has the disease. The hospital has only eight testing kits and has decided to pool blood samples of patients into eight vials for the tests. The patients are numbered $1$ through $16,$ and the vials are labelled $\text{A, B, C, D, E, F, G,}$ and $\text{H}.$ The following table shows the vials into which each patient’s blood sample is distributed.
| Patient |
Vials |
Patient |
Vials |
| 1 |
B,D,F,H |
9 |
A,D,F,H |
| 2 |
B,D,F,G |
10 |
A,D,F,G |
| 3 |
B,D,E,H |
11 |
A,D,E,H |
| 4 |
B,D,E,G |
12 |
A,D,E,G |
| 5 |
B,C,F,H |
13 |
A,C,F,H |
| 6 |
B,C,F,G |
14 |
A,C,F,G |
| 7 |
B,C,E,H |
15 |
A,C,E,H |
| 8 |
B,C,E,G |
16 |
A,C,E,G |
If a patient has the disease, then each vial containing his/her blood sample will test positive. If a vial tests positive, one of the patients whose blood samples were mixed in the vial has the disease. If a vial tests negative, then none of the patients whose blood samples were mixed in the vial has the disease.
Suppose one of the lab assistants accidentally mixed two patients’ blood samples before they were distributed to the vials. Which of the following correctly represents the set of all possible numbers of positive test results out of the eight vials $?$
- $ \{ 4,5 \} $
- $ \{ 5,6,7,8 \} $
- $ \{ 4,5,6,7,8 \} $
- $ \{ 4,5,6,7 \} $