A state planning commission reviewed 350 rural development grants to evaluate their alloca… — State PSC Exam

A state planning commission reviewed $350$ rural development grants to evaluate their allocation across three critical sectors: Healthcare ( $H$ ), Education ( $E$ ), and Infrastructure ( $I$ ). The review revealed the following data:

- $160$ grants were allocated to Healthcare.
- $150$ grants were allocated to Education.
- $140$ grants were allocated to Infrastructure.
- $50$ grants were allocated to both Healthcare and Education.
- $45$ grants were allocated to both Education and Infrastructure.
- $40$ grants were allocated to both Healthcare and Infrastructure.
- $15$ grants were allocated to all three sectors.

Based on the information above, which of the following statements are correct?

Exactly $20$ grants were not allocated to any of these three sectors.Answer
B
Exactly $135$ grants were allocated to exactly two sectors.
Exactly $70$ grants were allocated only to Education.Answer
D
Exactly $50$ grants were allocated to only Healthcare and Education.

Answer

The correct statements are that exactly 20 grants were not allocated to any of these three sectors, and exactly 70 grants were allocated only to Education.

The correct statements accurately identify the number of unallocated grants and the grants allocated exclusively to Education. The total number of allocated grants is found using the formula

|H \cup E \cup I| = 160 + 150 + 140 - 50 - 45 - 40 + 15 = 330

. Therefore,

350 - 330 = 20

grants are unallocated. Additionally, grants for ONLY Education equals the total for Education minus all its intersections:

150 - (35 + 30 + 15) = 70

Step-by-Step Solution

Identify the values for the intersections of exactly two sets by subtracting the 'all three' value (

15

) from the given two-set intersections.

Healthcare & Education only =

50 - 15 = 35

. Education & Infrastructure only =

45 - 15 = 30

. Healthcare & Infrastructure only =

40 - 15 = 25

The given intersections (e.g., Healthcare and Education) include the grants that belong to all three sectors. These must be separated to find exclusive regions.

Calculate the number of grants allocated to exactly two sectors.

Total for exactly two =

35 + 30 + 25 = 90

Adding the distinct 'only two' regions provides the total for grants overlapping in exactly two sectors.

Calculate the number of grants allocated exclusively to each single sector.

Only Healthcare =

160 - (35 + 25 + 15) = 85

. Only Education =

150 - (35 + 30 + 15) = 70

. Only Infrastructure =

140 - (25 + 30 + 15) = 70

Subtracting all overlapping parts of a set from its total yields the exclusive 'only one category' count.

Determine the total number of grants allocated to at least one sector and those allocated to none.

Total allocated =

85 + 70 + 70 + 35 + 30 + 25 + 15 = 330

. Unallocated (None) =

350 - 330 = 20

Summing all distinct regions of the Venn diagram gives the union of the sets. Subtracting this from the universal set gives the complement (None).

Evaluate each given statement based on the calculated regions.

Statement 1 (20 unallocated) is true. Statement 2 (135 exactly two) is false (actual is 90). Statement 3 (70 only Education) is true. Statement 4 (50 only H and E) is false (actual is 35).

Matching calculated values against the options reveals the correct selections.

Key Concept

Solving 3-set logical Venn diagrams using the inclusion-exclusion principle and identifying specific overlapping regions.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic