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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 11 through 1616, and the vials are labelled A, B, C, D, E, F, G, and H. The following table shows the vials into which each patient's blood sample is distributed.

PatientVialsPatientVials
1B, D, F, H9A, D, F, H
2B, D, F, G10A, D, F, G
3B, D, E, H11A, D, E, H
4B, D, E, G12A, D, E, G
5B, C, F, H13A, C, F, H
6B, C, F, G14A, C, F, G
7B, C, E, H15A, C, E, H
8B, C, E, G16A, 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?

Solution

✅ Correct Option: 3
Slide 1/2

Understanding the Set:

  1. This is a question-based set. In this set, you are told that there are 16 patients and 8 vials — labeled A, B, C, D, E, F, G, and H.

(Vials are simply the containers in which blood samples are stored.)

  1. It is also given that only one person has the disease.
  1. If a patient has the disease, then every vial containing his/her blood sample will test positive.

a) If a vial tests positive, at least one of the patients whose blood samples were mixed in that vial has the disease.

b) If a vial tests negative, then none of the patients whose samples were mixed in that vial has the disease.

This set might initially seem a bit confusing, but it was actually one of the simplest sets asked.

The key was to understand the passage correctly — once that was done, the questions were softballs.

PatientVialsPatientVials
1B, D, F, H9A, D, F, H
2B, D, F, G10A, D, F, G
3B, D, E, H11A, D, E, H
4B, D, E, G12A, D, E, G
5B, C, F, H13A, C, F, H
6B, C, F, G14A, C, F, G
7B, C, E, H15A, C, E, H
8B, C, E, G16A, C, E, G

Since every patient's blood sample is there in 44 vials

so with a mixing of two non-diseased patient's blood samples there will be 44 vials with positive test result.

Let one of the patients,

patient 1 or patient 16 has the disease and his blood is mixed with other them all 8 vials will tests positive. ⇒ 8 has to be one of the answers.

If patient 2 and patients 16’s blood is mixed of one of them has the disease then 7 of the 8 vials will test positive.

=> So 7 has to be there in the option.

If 1 has the disease and 1, 7 are mixed then 6 out the 8 vials tests positive.

IF 1 has the disease and 1,9 are mixed then 5 of the 8 vials tests positive,

Now, let us assume that patient 1 has the disease if his blood is not mixed, then 4 vials will definitely show positive.

So the correct answer is 3rd option

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