Question:medium

A survey of 600 schools in India was conducted to gather information about their online teaching learning processes (OTLP). The following four facilities were studied.
F1: Own software for OTLP
F2: Trained teachers for OTLP
F3: Training materials for OTLP
F4: All students having Laptops
The following observations were summarized from the survey.
1. 80 schools did not have any of the four facilities – F1, F2, F3, F4.
2. 40 schools had all four facilities.
3. The number of schools with only F1, only F2, only F3, and only F4 was 25, 30, 26 and 20 respectively.
4. The number of schools with exactly three of the facilities was the same irrespective of which three were considered.
5. 313 schools had F2.
6. 26 schools had only F2 and F3 (but neither F1 nor F4).
7. Among the schools having F4, 24 had only F3, and 45 had only F2.
8. 162 schools had both F1 and F2.
9. The number of schools having F1 was the same as the number of schools having F4.
What was the number of schools having only facilities F1 and F4? [This Question was asked as TITA]

Updated On: Jan 15, 2026
  • 20
  • 30
  • 10
  • 40
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The Correct Option is A

Solution and Explanation

To address the problem, we will determine the quantity of schools possessing exclusively facilities F1 and F4. This will be achieved through a step-by-step analysis employing set notation and Venn diagram principles:
  • Define the number of schools possessing:
    • F1 = A
    • F2 = B
    • F3 = C
    • F4 = D
  • Provided Data:
    • Total schools (T) = 600
    • Schools with no facilities = 80
    • Schools with all four facilities = 40
    • Schools with only:
      • F1 = 25
      • F2 = 30
      • F3 = 26
      • F4 = 20
    • Schools with F2 = 313
    • Schools with only F2 and F3 = 26
    • Schools with F4 and only F3 = 24
    • Schools with F4 and only F2 = 45
    • Schools with F1 and F2 = 162
    • A = D
  • Let:
    • a: represents schools with only F1 and F4 (our objective)
    • b: represents schools with only F1, F2, and F4
    • c: represents schools with only F1, F3, and F4
    • e: represents schools with only F2, F3, and F4
  • Given that the count of schools with exactly three facilities is constant, we establish:
    • x = represents the number of schools in any specific combination of three facilities
  • Formulated Equations:
    • A = a + 25 (only F1) + i (only F1 and F2) + l (only F1 and F3) + c (only F1, F3, and F4) + b (only F1, F2, and F4) + k (all four)
    • D = a + 20 (only F4) + k + 24 (only F3 and F4) + 45 (only F2 and F4) + c (only F1, F3, and F4) + b (only F1, F2, and F4)
  • Applying the given values to resolve the problem:
    • x = b = c = e = 10
    • Schools with no facilities = 80, thus T - 80 = 520 schools possess at least one facility.
    • Total for F1 = Total for F4
    • 520 = 25 + 30 + 26 + 20 + (3 * 10 for three-facility combinations) + 40 + a
    • 520 = 111 + 30 + 40 + a
    • 520 = 181 + a
    • a = 339
Consequently, the number of schools possessing only facilities F1 and F4 is 339.
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