Venn Diagrams and Set-Based Data

In a civil services training academy of $120$ officers, $70$ officers speak English, $60$ officers speak Hindi, and $30$ officers speak both English and Hindi. Based on this set data, which of the following statements are correct?

A district administration conducted a specialized training audit for a cadre of $300$ officers across three administrative domains: E-Governance ($E$), Disaster Management ($D$), and Financial Administration ($F$). The data gathered from the audit is as follows:

- $140$ officers are trained in E-Governance.
- $130$ officers are trained in Disaster Management.
- $120$ officers are trained in Financial Administration.
- $50$ officers are trained in both E-Governance and Disaster Management.
- $40$ officers are trained in both Disaster Management and Financial Administration.
- $45$ officers are trained in both E-Governance and Financial Administration.
- $20$ officers are trained in all three domains.

Based on the data provided, which of the following statements are correct?

In a rural development assessment of $200$ villages in a district, data regarding access to three basic infrastructure facilities—Piped Water, Solar Power, and Broadband Internet—was recorded as follows:
- $110$ villages have access to Piped Water.
- $100$ villages have access to Solar Power.
- $90$ villages have access to Broadband Internet.
- $50$ villages have access to both Piped Water and Solar Power.
- $40$ villages have access to both Solar Power and Broadband Internet.
- $45$ villages have access to both Piped Water and Broadband Internet.
- $20$ villages have access to all three facilities.

Which of the following statements are correct?

In a state administrative academy, a batch of $300$ probationary officers were surveyed regarding their enrolment in three specialized training modules: Cyber Security ($C$), Public Policy ($P$), and Financial Management ($F$). The survey revealed the following data:
- Total officers enrolled in Cyber Security: $160$
- Total officers enrolled in Public Policy: $140$
- Total officers enrolled in Financial Management: $130$
- Officers enrolled in both Cyber Security and Public Policy: $65$
- Officers enrolled in both Public Policy and Financial Management: $55$
- Officers enrolled in both Cyber Security and Financial Management: $50$
- Officers enrolled in all three modules: $20$

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

In a administrative evaluation conducted for $250$ government officers, data was collected regarding the completion of three specialized training modules: Cyber Security ($C$), Data Analytics ($D$), and Public Procurement ($P$).

- $130$ officers completed Cyber Security
- $110$ officers completed Data Analytics
- $100$ officers completed Public Procurement
- $55$ officers completed both Cyber Security and Data Analytics
- $40$ officers completed both Data Analytics and Public Procurement
- $45$ officers completed both Cyber Security and Public Procurement
- $20$ officers completed all three modules

Based on the information provided, how many officers completed exactly two training modules?

A survey was conducted among $400$ administrative officers in a state secretariat regarding their operational oversight of three major public welfare projects: Project Alpha, Project Beta, and Project Gamma. The survey collected the following data:

- $190$ officers oversee Project Alpha.
- $180$ officers oversee Project Beta.
- $170$ officers oversee Project Gamma.
- $85$ officers oversee both Project Alpha and Project Beta.
- $75$ officers oversee both Project Beta and Project Gamma.
- $70$ officers oversee both Project Alpha and Project Gamma.
- $40$ officers do not oversee any of these three projects.

Based on the given information, how many administrative officers oversee exactly two projects?

During an urban mobility audit conducted among $400$ daily commuters in a metropolitan city, data was recorded regarding their regular use of three transit modes: Metro Rail ($M$), Electric Bus ($E$), and Shared Bicycle ($B$). The survey revealed that $190$ commuters use Metro Rail, $160$ use Electric Bus, and $120$ use Shared Bicycle. Furthermore, $60$ commuters use both Metro Rail and Electric Bus, $40$ use both Electric Bus and Shared Bicycle, and $50$ use both Metro Rail and Shared Bicycle. If $20$ commuters utilize all three modes of transport, how many commuters do not use any of these three transit modes?

A survey was conducted among $500$ rural households in a district regarding their adoption of three e-governance services: Digital Payments ($P$), E-Health Cards ($H$), and Online Land Records ($L$). The survey revealed that $210$ households use Digital Payments, $190$ use E-Health Cards, and $220$ use Online Land Records. Additionally, $80$ households use both Digital Payments and E-Health Cards, $70$ use both E-Health Cards and Online Land Records, and $90$ use both Digital Payments and Online Land Records. If $40$ households use all three e-governance services, how many households use at least two of these services?

In a performance audit of $200$ public health centers across a administrative division, adoption of three digital portals was evaluated: Tele-consultation ($T$), Electronic Health Records ($E$), and Medicine Inventory Management ($M$). The audit revealed the following findings:

- $105$ centers use Tele-consultation ($T$)
- $90$ centers use Electronic Health Records ($E$)
- $85$ centers use Medicine Inventory Management ($M$)
- $40$ centers use both $T$ and $E$
- $35$ centers use both $E$ and $M$
- $30$ centers use both $T$ and $M$
- $15$ centers use all three platforms

How many public health centers utilize exactly two of the three digital health platforms?

In an agricultural extension survey conducted among $250$ farmers in an administrative block, awareness of three welfare schemes was evaluated: Crop Insurance ($C$), Soil Health Card ($S$), and Kisan Credit Card ($K$). The survey revealed that $120$ farmers are aware of $C$, $110$ are aware of $S$, and $130$ are aware of $K$. Additionally, $45$ farmers are aware of both $C$ and $S$, $50$ are aware of both $S$ and $K$, and $40$ are aware of both $C$ and $K$. If $15$ farmers are aware of all three schemes, how many farmers are aware of exactly two of these schemes?

A survey was conducted among $500$ civil service aspirants in a coaching institute regarding their daily newspaper reading habits among three publications: *The Hindu*, *The Indian Express*, and *Business Standard*.

The survey revealed the following data:
- The total number of aspirants reading *The Hindu*, *The Indian Express*, and *Business Standard* are $240$, $210$, and $180$ respectively.
- Exactly $20$ aspirants read all three newspapers.
- The ratio of the number of aspirants who read ONLY *The Hindu* and *The Indian Express* to those who read ONLY *The Indian Express* and *Business Standard* to those who read ONLY *The Hindu* and *Business Standard* is $3 : 2 : 4$.
- The number of aspirants who read ONLY *Business Standard* is $70$.
- The number of aspirants who read ONLY *The Hindu* is equal to the number of aspirants who read ONLY *The Indian Express*.

Based on the data provided, how many aspirants in total do NOT read any of the three newspapers?

In a regional administrative office of $80$ officers, $45$ officers completed a training course in Data Analytics, $40$ officers completed a course in Public Policy, and $15$ officers completed both training courses. How many officers completed neither of the two training courses?

In a survey of $100$ residents in a locality, $60$ residents read Newspaper X, $50$ read Newspaper Y, and $20$ read both Newspaper X and Newspaper Y. How many residents read neither Newspaper X nor Newspaper Y?

In a survey of $120$ civil service aspirants, $65$ read Newspaper A, $55$ read Newspaper B, and $45$ read Newspaper C. Additionally, $25$ read both A and B, $20$ read both B and C, $15$ read both A and C, and $8$ read all three newspapers. How many aspirants read exactly two of these newspapers?

In a survey of $500$ civil service aspirants preparing for State PSC examinations regarding their daily newspaper reading habits:
- $260$ aspirants read Newspaper A
- $220$ aspirants read Newspaper B
- $180$ aspirants read Newspaper C
- $90$ aspirants read both Newspaper A and Newspaper B
- $70$ aspirants read both Newspaper B and Newspaper C
- $80$ aspirants read both Newspaper A and Newspaper C
- $30$ aspirants read all three newspapers

How many aspirants read exactly one of these three newspapers?

A survey was conducted among $300$ State PSC aspirants regarding their preparation for three subjects: General Studies ($GS$), General Aptitude Test ($C$), and Optional Subject ($O$). The data collected is as follows:
- $180$ candidates prepare for $GS$
- $140$ candidates prepare for $C$
- $120$ candidates prepare for $O$
- $80$ candidates prepare for both $GS$ and $C$
- $50$ candidates prepare for both $C$ and $O$
- $60$ candidates prepare for both $GS$ and $O$
- $30$ candidates prepare for all three subjects

Which of the following statements are correct? (Select all correct statements)