{"id":89928,"date":"2025-06-01T10:17:24","date_gmt":"2025-06-01T10:17:24","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=89928"},"modified":"2025-06-01T10:17:24","modified_gmt":"2025-06-01T10:17:24","slug":"in-an-examination-there-are-three-subjects-a-b-and-c-a-studen","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/","title":{"rendered":"In an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A studen"},"content":{"rendered":"

In an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A student has to pass in each subject. 20% students failed in ‘A’, 22% students failed in ‘B’ and 16% students failed in ‘C’. The total number of students passing the whole examination lies between :<\/p>\n

[amp_mcq option1=”42% and 84%” option2=”42% and 78%” option3=”58% and 78%” option4=”58% and 84%” correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2015<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%.
\n<\/section>\n
\nLet F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X.
\nF(A) = 20%, P(A) = 100 – 20 = 80%
\nF(B) = 22%, P(B) = 100 – 22 = 78%
\nF(C) = 16%, P(C) = 100 – 16 = 84%
\nA student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C).
\nMaximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%.\nMinimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap).\nMax P(F(A U B U C)) <= F(A) + F(B) + F(C) = 20 + 22 + 16 = 58%.\nThe minimum percentage passing all = 100% - Max P(F(A U B U C)) = 100% - 58% = 42%.\nAlternatively, using the inclusion-exclusion principle for passing percentages, minimum P(A \u2229 B \u2229 C) >= P(A) + P(B) + P(C) – 2 * 100% = 80 + 78 + 84 – 200 = 242 – 200 = 42%.
\nThus, the total number of students passing the whole examination lies between 42% and 78%.
\n<\/section>\n
\nThe range calculated represents the theoretical bounds based on the given failure percentages. The maximum occurs when failures are mutually exclusive, leading to minimal success overlap. The minimum occurs when failures are maximally overlapping, leading to maximal success overlap (specifically, the students failing at least one subject are minimized, which happens when they overlap maximally, pushing the ‘pass all’ group down). The minimum passing percentage calculation using the sum of passing percentages minus 200% is a quick formula derived from inclusion-exclusion for three sets.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

In an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A student has to pass in each subject. 20% students failed in ‘A’, 22% students failed in ‘B’ and 16% students failed in ‘C’. The total number of students passing the whole examination lies between : [amp_mcq option1=”42% and 84%” option2=”42% and 78%” option3=”58% … <\/p>\n

Detailed SolutionIn an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A studen<\/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":[1085],"tags":[1443,1102],"class_list":["post-89928","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1443","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIn an examination, there are three subjects 'A', 'B' and 'C'. A studen<\/title>\n<meta name=\"description\" content=\"The correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%. Let F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X. F(A) = 20%, P(A) = 100 - 20 = 80% F(B) = 22%, P(B) = 100 - 22 = 78% F(C) = 16%, P(C) = 100 - 16 = 84% A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C). Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) = P(A) + P(B) + P(C) - 2 * 100% = 80 + 78 + 84 - 200 = 242 - 200 = 42%. Thus, the total number of students passing the whole examination lies between 42% and 78%.\" \/>\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-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In an examination, there are three subjects 'A', 'B' and 'C'. A studen\" \/>\n<meta property=\"og:description\" content=\"The correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%. Let F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X. F(A) = 20%, P(A) = 100 - 20 = 80% F(B) = 22%, P(B) = 100 - 22 = 78% F(C) = 16%, P(C) = 100 - 16 = 84% A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C). Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) = P(A) + P(B) + P(C) - 2 * 100% = 80 + 78 + 84 - 200 = 242 - 200 = 42%. Thus, the total number of students passing the whole examination lies between 42% and 78%.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:17:24+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=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"In an examination, there are three subjects 'A', 'B' and 'C'. A studen","description":"The correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%. Let F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X. F(A) = 20%, P(A) = 100 - 20 = 80% F(B) = 22%, P(B) = 100 - 22 = 78% F(C) = 16%, P(C) = 100 - 16 = 84% A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C). Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) = P(A) + P(B) + P(C) - 2 * 100% = 80 + 78 + 84 - 200 = 242 - 200 = 42%. Thus, the total number of students passing the whole examination lies between 42% and 78%.","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-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/","og_locale":"en_US","og_type":"article","og_title":"In an examination, there are three subjects 'A', 'B' and 'C'. A studen","og_description":"The correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%. Let F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X. F(A) = 20%, P(A) = 100 - 20 = 80% F(B) = 22%, P(B) = 100 - 22 = 78% F(C) = 16%, P(C) = 100 - 16 = 84% A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C). Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) = P(A) + P(B) + P(C) - 2 * 100% = 80 + 78 + 84 - 200 = 242 - 200 = 42%. Thus, the total number of students passing the whole examination lies between 42% and 78%.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:17:24+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/","url":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/","name":"In an examination, there are three subjects 'A', 'B' and 'C'. A studen","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:17:24+00:00","dateModified":"2025-06-01T10:17:24+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is B. The percentage of students passing all three subjects must lie between 42% and 78%. Let F(X) be the percentage of students failing subject X, and P(X) be the percentage passing subject X. F(A) = 20%, P(A) = 100 - 20 = 80% F(B) = 22%, P(B) = 100 - 22 = 78% F(C) = 16%, P(C) = 100 - 16 = 84% A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A \u2229 B \u2229 C). Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A \u2229 B \u2229 C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) = P(A) + P(B) + P(C) - 2 * 100% = 80 + 78 + 84 - 200 = 242 - 200 = 42%. Thus, the total number of students passing the whole examination lies between 42% and 78%.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/in-an-examination-there-are-three-subjects-a-b-and-c-a-studen\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"In an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A studen"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89928","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=89928"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89928\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=89928"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=89928"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=89928"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}