Question

In a survey of 100 out-patients who reported at a hospital on a day, it was found that 70 complained of fever, 50 had stomach troubles and 30 were injured. The entire 100 patients had one or more of the complaints, 44 had exactly two of them.

How many patients had all the three complaints?

Answer 1

3 patients had all the three complaints

Explanation:

Let U be the set of patients who reported at the hospital on that day

Let F be the set of patients who complained of fever

Let S be the set of patients who had stomach troubles

Let I be the set of injured patients

Then the given data can be written as:

• n(U) = n(F∪S∪I) = 100
• n(F) = 70
• n(S) = 50
• n(I) = 30
• n(F∩S) + n(S∩I) + n(I∩F) - 3×n(F∩S∩I) = 44

n(F∩S∩I) = ?

Using the formula for the cardinal number of union of three sets:

n(F∪S∪I) = n(F) + n(S) + n(I) - n(F∩S) - n(S∩I) - n(I∩F) + n(F∩S∩I)

100 = 70 + 50 + 30 - (44 + 3×n(F∩S∩I)) + n(F∩S∩I)

100 = 150 - 44 - 2×n(F∩S∩I)

2×n(F∩S∩I) = 106 - 100 = 6

n(F∩S∩I) = 3