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Difficulty: MediumVenn Diagrams and Set-Based Data

A survey was conducted among 500500 rural households in a district regarding their adoption of three e-governance services: Digital Payments (PP), E-Health Cards (HH), and Online Land Records (LL). The survey revealed that 210210 households use Digital Payments, 190190 use E-Health Cards, and 220220 use Online Land Records. Additionally, 8080 households use both Digital Payments and E-Health Cards, 7070 use both E-Health Cards and Online Land Records, and 9090 use both Digital Payments and Online Land Records. If 4040 households use all three e-governance services, how many households use at least two of these services?

  1. A
    120120
  2. 160160Answer
  3. C
    200200
  4. D
    240240

Answer

The number of households using at least two e-governance services is 160160.
To find the number of households using at least two services, we sum the households using exactly two services and those using all three services. The households using only Digital Payments and E-Health Cards is 80−40=4080 - 40 = 40. The households using only E-Health Cards and Online Land Records is 70−40=3070 - 40 = 30. The households using only Digital Payments and Online Land Records is 90−40=5090 - 40 = 50. Adding these to the 4040 households using all three services yields 40+30+50+40=16040 + 30 + 50 + 40 = 160.

Step-by-Step Solution

1
Identify given set parameters and triple intersection.
Total N=500N = 500, ∣P∩H∣=80|P \cap H| = 80, ∣H∩L∣=70|H \cap L| = 70, ∣P∩L∣=90|P \cap L| = 90, and ∣P∩H∩L∣=40|P \cap H \cap L| = 40.
Break down pairwise intersections into mutually exclusive regions.
2
Calculate households using exactly two services for each pair.
Only PP and H=80−40=40H = 80 - 40 = 40; Only HH and L=70−40=30L = 70 - 40 = 30; Only PP and L=90−40=50L = 90 - 40 = 50.
Subtract the triple intersection count (4040) from each pairwise intersection to isolate households using exactly two services.
3
Sum households using exactly two services and households using all three services.
At least two services =40+30+50+40=160= 40 + 30 + 50 + 40 = 160.
'At least two' encompasses both 'exactly two' and 'all three' regions of the Venn diagram.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Mutually Exclusive Region Calculation
Estimated Time:1m 30s
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Topics

Venn Diagrams and Set-Based Data

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