{"id":93224,"date":"2025-06-01T11:45:20","date_gmt":"2025-06-01T11:45:20","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=93224"},"modified":"2025-06-01T11:45:20","modified_gmt":"2025-06-01T11:45:20","slug":"in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/","title":{"rendered":"In a party, each person takes at least one beverage. There are three b"},"content":{"rendered":"

In a party, each person takes at least one beverage. There are three beverages in the party \u2013 tea, coffee and milk. Each beverage is consumed by 30 persons. Five persons, who take tea, also take milk. Ten persons, who take milk, also take coffee. However, no person takes tea and coffee both. How many persons are there in the party ?<\/p>\n

[amp_mcq option1=”90″ option2=”85″ option3=”80″ option4=”75″ correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CISF-AC-EXE – 2023<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption.
\n<\/section>\n
\n– Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively.
\n– We are given: |T| = 30, |C| = 30, |M| = 30.
\n– |T \u2229 M| = 5 (Persons who take tea and milk).
\n– |M \u2229 C| = 10 (Persons who take milk and coffee).
\n– |T \u2229 C| = 0 (No person takes tea and coffee both).
\n– Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0).
\n– Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|.
\n– The Principle of Inclusion-Exclusion for three sets is:
\n |T \u222a C \u222a M| = |T| + |C| + |M| – (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M|
\n– Substitute the given values:
\n |T \u222a C \u222a M| = 30 + 30 + 30 – (0 + 5 + 10) + 0
\n |T \u222a C \u222a M| = 90 – 15 + 0
\n |T \u222a C \u222a M| = 75.
\n<\/section>\n
\nThis problem can also be visualized using a Venn diagram. Starting with the intersections, we know |T \u2229 C| = 0. Given |T \u2229 M| = 5 and |M \u2229 C| = 10, and |T \u2229 C \u2229 M| = 0, the people taking exactly two beverages are:
\n– Tea and Milk only: 5 – 0 = 5
\n– Milk and Coffee only: 10 – 0 = 10
\n– Tea and Coffee only: 0
\nPeople taking only one beverage:
\n– Tea only: |T| – (|T \u2229 M| + |T \u2229 C|) + |T \u2229 C \u2229 M| = 30 – (5 + 0) + 0 = 25
\n– Coffee only: |C| – (|C \u2229 M| + |C \u2229 T|) + |T \u2229 C \u2229 M| = 30 – (10 + 0) + 0 = 20
\n– Milk only: |M| – (|M \u2229 T| + |M \u2229 C|) + |T \u2229 C \u2229 M| = 30 – (5 + 10) + 0 = 15
\nTotal persons = (Tea only) + (Coffee only) + (Milk only) + (Tea & Milk only) + (Milk & Coffee only) + (Tea & Coffee only) + (All three)
\nTotal = 25 + 20 + 15 + 5 + 10 + 0 + 0 = 75.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

In a party, each person takes at least one beverage. There are three beverages in the party \u2013 tea, coffee and milk. Each beverage is consumed by 30 persons. Five persons, who take tea, also take milk. Ten persons, who take milk, also take coffee. However, no person takes tea and coffee both. How many … <\/p>\n

Detailed SolutionIn a party, each person takes at least one beverage. There are three b<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1089],"tags":[1105,1102],"class_list":["post-93224","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1105","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIn a party, each person takes at least one beverage. There are three b<\/title>\n<meta name=\"description\" content=\"The correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption. - Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively. - We are given: |T| = 30, |C| = 30, |M| = 30. - |T \u2229 M| = 5 (Persons who take tea and milk). - |M \u2229 C| = 10 (Persons who take milk and coffee). - |T \u2229 C| = 0 (No person takes tea and coffee both). - Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0). - Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|. - The Principle of Inclusion-Exclusion for three sets is: |T \u222a C \u222a M| = |T| + |C| + |M| - (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M| - Substitute the given values: |T \u222a C \u222a M| = 30 + 30 + 30 - (0 + 5 + 10) + 0 |T \u222a C \u222a M| = 90 - 15 + 0 |T \u222a C \u222a M| = 75.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In a party, each person takes at least one beverage. There are three b\" \/>\n<meta property=\"og:description\" content=\"The correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption. - Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively. - We are given: |T| = 30, |C| = 30, |M| = 30. - |T \u2229 M| = 5 (Persons who take tea and milk). - |M \u2229 C| = 10 (Persons who take milk and coffee). - |T \u2229 C| = 0 (No person takes tea and coffee both). - Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0). - Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|. - The Principle of Inclusion-Exclusion for three sets is: |T \u222a C \u222a M| = |T| + |C| + |M| - (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M| - Substitute the given values: |T \u222a C \u222a M| = 30 + 30 + 30 - (0 + 5 + 10) + 0 |T \u222a C \u222a M| = 90 - 15 + 0 |T \u222a C \u222a M| = 75.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:45:20+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"In a party, each person takes at least one beverage. There are three b","description":"The correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption. - Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively. - We are given: |T| = 30, |C| = 30, |M| = 30. - |T \u2229 M| = 5 (Persons who take tea and milk). - |M \u2229 C| = 10 (Persons who take milk and coffee). - |T \u2229 C| = 0 (No person takes tea and coffee both). - Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0). - Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|. - The Principle of Inclusion-Exclusion for three sets is: |T \u222a C \u222a M| = |T| + |C| + |M| - (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M| - Substitute the given values: |T \u222a C \u222a M| = 30 + 30 + 30 - (0 + 5 + 10) + 0 |T \u222a C \u222a M| = 90 - 15 + 0 |T \u222a C \u222a M| = 75.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/","og_locale":"en_US","og_type":"article","og_title":"In a party, each person takes at least one beverage. There are three b","og_description":"The correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption. - Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively. - We are given: |T| = 30, |C| = 30, |M| = 30. - |T \u2229 M| = 5 (Persons who take tea and milk). - |M \u2229 C| = 10 (Persons who take milk and coffee). - |T \u2229 C| = 0 (No person takes tea and coffee both). - Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0). - Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|. - The Principle of Inclusion-Exclusion for three sets is: |T \u222a C \u222a M| = |T| + |C| + |M| - (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M| - Substitute the given values: |T \u222a C \u222a M| = 30 + 30 + 30 - (0 + 5 + 10) + 0 |T \u222a C \u222a M| = 90 - 15 + 0 |T \u222a C \u222a M| = 75.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:45:20+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/","url":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/","name":"In a party, each person takes at least one beverage. There are three b","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:45:20+00:00","dateModified":"2025-06-01T11:45:20+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is 75 persons. This can be determined using the principle of inclusion-exclusion for sets, given the information about beverage consumption. - Let T, C, and M represent the sets of people who take Tea, Coffee, and Milk, respectively. - We are given: |T| = 30, |C| = 30, |M| = 30. - |T \u2229 M| = 5 (Persons who take tea and milk). - |M \u2229 C| = 10 (Persons who take milk and coffee). - |T \u2229 C| = 0 (No person takes tea and coffee both). - Since no one takes tea and coffee together (|T \u2229 C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T \u2229 C \u2229 M| = 0). - Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T \u222a C \u222a M|. - The Principle of Inclusion-Exclusion for three sets is: |T \u222a C \u222a M| = |T| + |C| + |M| - (|T \u2229 C| + |T \u2229 M| + |C \u2229 M|) + |T \u2229 C \u2229 M| - Substitute the given values: |T \u222a C \u222a M| = 30 + 30 + 30 - (0 + 5 + 10) + 0 |T \u222a C \u222a M| = 90 - 15 + 0 |T \u222a C \u222a M| = 75.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/in-a-party-each-person-takes-at-least-one-beverage-there-are-three-b\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CISF-AC-EXE","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cisf-ac-exe\/"},{"@type":"ListItem","position":3,"name":"In a party, each person takes at least one beverage. 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