Logical Venn Diagrams

An environmental agency reviewed the compliance of $380$ industrial plants across three pollution control standards: Air Emissions ($A$), Water Discharge ($W$), and Solid Waste ($S$). The review revealed the following data:
- $170$ plants met standard $A$.
- $150$ plants met standard $W$.
- $160$ plants met standard $S$.
- $60$ plants met both $A$ and $W$.
- $55$ plants met both $W$ and $S$.
- $50$ plants met both $A$ and $S$.
- $20$ plants failed to meet any of the three standards.

Based on this information, how many industrial plants met EXACTLY TWO of the pollution control standards?

A state health department surveyed $800$ rural clinics regarding the availability of three essential facilities: Telemedicine ($T$), Maternal Care Units ($M$), and $24/7$ Ambulance Services ($A$).

The survey results revealed the following:
- $410$ clinics have Telemedicine.
- $460$ clinics have Maternal Care Units.
- $400$ clinics have $24/7$ Ambulance Services.
- $220$ clinics have both Telemedicine and Maternal Care Units.
- $230$ clinics have both Maternal Care Units and Ambulance Services.
- $210$ clinics have both Telemedicine and Ambulance Services.
- $50$ clinics do not have any of these three facilities.

Based on the data provided, which of the following statements are correct? (Select all that apply)

A city library conducted a reading preferences survey among $600$ of its registered members. The survey focused on three genres: Mystery ($M$), Sci-Fi ($S$), and Biography ($B$). The results showed that $120$ members do not read any of these three genres.

Furthermore, the survey revealed the following:
- $250$ members read Mystery.
- $210$ members read Sci-Fi.
- $180$ members read Biography.
- $70$ members read both Mystery and Biography.
- $60$ members read both Sci-Fi and Biography.
- $40$ members read all three genres.

Based on the survey results, how many members read both Mystery and Sci-Fi, but NOT Biography?

In a regional sports academy, a group of $70$ trainees were surveyed about their participation in two sports: Cricket and Football. $45$ trainees play Cricket, $30$ trainees play Football, and $10$ trainees play neither sport. How many trainees play both Cricket and Football?

In a state policy research institute, a survey of $200$ research analysts was conducted regarding their domain specializations in three sectors: Health ($H$), Education ($E$), and Agriculture ($A$). The data revealed that $90$ analysts specialize in Health, $85$ in Education, and $80$ in Agriculture. Furthermore, $35$ analysts specialize in both Health and Education, $30$ in both Education and Agriculture, $25$ in both Health and Agriculture, and $10$ analysts specialize in all three sectors. How many research analysts specialize in corporate policy research analysts who specialize in exactly two of these sectors?

In a technology seminar attended by $60$ software engineers, $35$ engineers know Python and $25$ engineers know Java. If $10$ engineers know both Python and Java, how many engineers know neither of these two programming languages?

In a regional agricultural department, a survey was conducted among $160$ field officers regarding their supervisory duties across three major crop initiatives: Wheat ($W$), Rice ($R$), and Cotton ($C$). The survey revealed that $85$ officers supervise Wheat, $70$ supervise Rice, and $60$ supervise Cotton. Furthermore, $35$ officers supervise both Wheat and Rice, $25$ supervise both Rice and Cotton, $30$ supervise both Wheat and Cotton, and $15$ officers supervise all three crop initiatives. How many field officers supervise exactly two of these three crop initiatives?

In a district administrative audit of $240$ agricultural cooperatives, compliance with three digital platforms was surveyed: E-Farming Portal ($P$), Soil Health Database ($S$), and Crop Insurance Registry ($I$). It was found that $115$ cooperatives use $P$, $110$ use $S$, and $105$ use $I$. Exactly $25$ cooperatives do not use any of the three platforms, and $20$ cooperatives use all three platforms. Furthermore, the number of cooperatives using only $P$, only $S$, and only $I$ are in the ratio $8 : 9 : 7$, respectively. How many cooperatives use exactly two of these digital platforms?

In a district administration department, a survey was conducted among $250$ officers regarding their technical expertise in three digital governance domains: Cyber Security ($C$), Data Analytics ($D$), and Public Grievance Portals ($P$). The data collected is as follows:
- $120$ officers have expertise in Data Analytics.
- $115$ officers have expertise in Cyber Security.
- $100$ officers have expertise in Public Grievance Portals.
- $50$ officers have expertise in both Data Analytics and Cyber Security.
- $45$ officers have expertise in both Cyber Security and Public Grievance Portals.
- $40$ officers have expertise in both Data Analytics and Public Grievance Portals.
- $20$ officers possess expertise in all three digital governance domains.

Based on the given data, how many officers have expertise in EXACTLY TWO of these three domains?

In a district administration office, a survey of $120$ officers was conducted regarding their proficiency in using three government portals: e-Office ($E$), CPGRAMS ($C$), and GeM ($G$). The survey revealed that $65$ officers are proficient in e-Office, $55$ in CPGRAMS, and $50$ in GeM. Furthermore, $25$ officers are proficient in both e-Office and CPGRAMS, $20$ in both CPGRAMS and GeM, and $22$ in both e-Office and GeM. If $10$ officers are proficient in all three portals, how many officers are proficient in none of these three portals?

In a state administrative audit of $300$ urban development projects, participation across three infrastructure domains—Sanitation ($S$), Transport ($T$), and Green Space ($G$)—was analyzed. The audit revealed that $145$ projects involve Sanitation, $135$ involve Transport, and $125$ involve Green Space. Exactly $70$ projects involve precisely two domains, while $20$ projects involve all three domains. Furthermore, $45$ projects involve both Sanitation and Transport, and $35$ projects involve both Transport and Green Space. Which of the following statements are correct? (Select all correct options.)

In a survey conducted among $300$ State PSC aspirants, data was collected regarding their enrollment in preparation modules for three optional subjects: Public Administration ($P$), Sociology ($S$), and Geography ($G$). It was found that $140$ aspirants enrolled in Public Administration, $130$ in Sociology, and $120$ in Geography. Further, $50$ aspirants enrolled in both Public Administration and Sociology, $45$ in both Sociology and Geography, and $40$ in both Public Administration and Geography. If $20$ aspirants did not enroll in any of these three subjects, how many aspirants enrolled in EXACTLY ONE optional subject?

In a group of $60$ civil service aspirants, $35$ aspirants study History, $25$ study Geography, and $10$ study both History and Geography. How many aspirants study neither History nor Geography?

In a survey of $120$ civil service aspirants, $65$ study Indian History, $55$ study Indian Polity, and $45$ study Geography. Further details reveal that $25$ aspirants study both History and Polity, $20$ study both Polity and Geography, $15$ study both History and Geography, and $8$ study all three subjects. How many aspirants study exactly two subjects?

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?

In a municipal office survey of $100$ administrative officers, $50$ officers speak English, $40$ officers speak Hindi, and $20$ officers speak both English and Hindi. Based on this information, which of the following statements are correct?

In a municipal department of $120$ officers, a survey was conducted regarding three operational responsibilities: Urban Planning ($U$), Environmental Compliance ($E$), and Public Works ($P$). The survey revealed the following data:
- $65$ officers handle Urban Planning ($U$).
- $50$ officers handle Environmental Compliance ($E$).
- $55$ officers handle Public Works ($P$).
- $5$ officers handle none of these three responsibilities.
- $45$ officers handle at least two of these responsibilities.
- $10$ officers handle all three responsibilities.
- $20$ officers handle both Urban Planning and Environmental Compliance.
- $18$ officers handle both Environmental Compliance and Public Works.

How many officers handle ONLY Urban Planning?