{"id":93120,"date":"2025-06-01T11:42:16","date_gmt":"2025-06-01T11:42:16","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=93120"},"modified":"2025-06-01T11:42:16","modified_gmt":"2025-06-01T11:42:16","slug":"in-a-group-of-30-students-each-student-has-opted-for-at-least-one-of","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/in-a-group-of-30-students-each-student-has-opted-for-at-least-one-of\/","title":{"rendered":"In a group of 30 students, each student has opted for at least one of"},"content":{"rendered":"

In a group of 30 students, each student has opted for at least one of the two subjects, Hindi and English. Twelve of them have opted for Hindi and twenty-two have opted for English. The number of students who have opted for only English is :<\/p>\n

[amp_mcq option1=”22″ option2=”18″ option3=”12″ option4=”8″ correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CISF-AC-EXE – 2022<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nLet H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H \\cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H \\cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H \\cup E| = |H| + |E| – |H \\cap E|$. Substituting the given values, $30 = 12 + 22 – |H \\cap E|$. This gives $30 = 34 – |H \\cap E|$, so $|H \\cap E| = 34 – 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. So, number of students who opted for only English = $|E| – |H \\cap E| = 22 – 4 = 18$.
\n<\/section>\n
\n– Total students = Students who opted for Hindi only + Students who opted for English only + Students who opted for both.
\n– $|H \\cup E| = |H \\text{ only}| + |E \\text{ only}| + |H \\cap E|$.
\n– $|H| = |H \\text{ only}| + |H \\cap E|$.
\n– $|E| = |E \\text{ only}| + |H \\cap E|$.
\n<\/section>\n
\nFrom $|H \\cap E|=4$, we can also find the number of students who opted for only Hindi: $|H \\text{ only}| = |H| – |H \\cap E| = 12 – 4 = 8$.
\nCheck: Total students = (Only Hindi) + (Only English) + (Both) = 8 + 18 + 4 = 30, which matches the given information.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

In a group of 30 students, each student has opted for at least one of the two subjects, Hindi and English. Twelve of them have opted for Hindi and twenty-two have opted for English. The number of students who have opted for only English is : [amp_mcq option1=”22″ option2=”18″ option3=”12″ option4=”8″ correct=”option2″] This question was … <\/p>\n

Detailed SolutionIn a group of 30 students, each student has opted for at least one of<\/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":[1108,1102],"class_list":["post-93120","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1108","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIn a group of 30 students, each student has opted for at least one of<\/title>\n<meta name=\"description\" content=\"Let H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H cup E| = |H| + |E| - |H cap E|$. Substituting the given values, $30 = 12 + 22 - |H cap E|$. This gives $30 = 34 - |H cap E|$, so $|H cap E| = 34 - 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. So, number of students who opted for only English = $|E| - |H cap E| = 22 - 4 = 18$. - Total students = Students who opted for Hindi only + Students who opted for English only + Students who opted for both. - $|H cup E| = |H text{ only}| + |E text{ only}| + |H cap E|$. - $|H| = |H text{ only}| + |H cap E|$. - $|E| = |E text{ only}| + |H cap E|$.\" \/>\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-group-of-30-students-each-student-has-opted-for-at-least-one-of\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In a group of 30 students, each student has opted for at least one of\" \/>\n<meta property=\"og:description\" content=\"Let H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H cup E| = |H| + |E| - |H cap E|$. Substituting the given values, $30 = 12 + 22 - |H cap E|$. This gives $30 = 34 - |H cap E|$, so $|H cap E| = 34 - 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. So, number of students who opted for only English = $|E| - |H cap E| = 22 - 4 = 18$. - Total students = Students who opted for Hindi only + Students who opted for English only + Students who opted for both. - $|H cup E| = |H text{ only}| + |E text{ only}| + |H cap E|$. - $|H| = |H text{ only}| + |H cap E|$. - $|E| = |E text{ only}| + |H cap E|$.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/in-a-group-of-30-students-each-student-has-opted-for-at-least-one-of\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:42:16+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 a group of 30 students, each student has opted for at least one of","description":"Let H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H cup E| = |H| + |E| - |H cap E|$. Substituting the given values, $30 = 12 + 22 - |H cap E|$. This gives $30 = 34 - |H cap E|$, so $|H cap E| = 34 - 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. So, number of students who opted for only English = $|E| - |H cap E| = 22 - 4 = 18$. - Total students = Students who opted for Hindi only + Students who opted for English only + Students who opted for both. - $|H cup E| = |H text{ only}| + |E text{ only}| + |H cap E|$. - $|H| = |H text{ only}| + |H cap E|$. - $|E| = |E text{ only}| + |H cap E|$.","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-group-of-30-students-each-student-has-opted-for-at-least-one-of\/","og_locale":"en_US","og_type":"article","og_title":"In a group of 30 students, each student has opted for at least one of","og_description":"Let H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H cup E| = |H| + |E| - |H cap E|$. Substituting the given values, $30 = 12 + 22 - |H cap E|$. This gives $30 = 34 - |H cap E|$, so $|H cap E| = 34 - 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. So, number of students who opted for only English = $|E| - |H cap E| = 22 - 4 = 18$. - Total students = Students who opted for Hindi only + Students who opted for English only + Students who opted for both. - $|H cup E| = |H text{ only}| + |E text{ only}| + |H cap E|$. - $|H| = |H text{ only}| + |H cap E|$. - $|E| = |E text{ only}| + |H cap E|$.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/in-a-group-of-30-students-each-student-has-opted-for-at-least-one-of\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:42:16+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-a-group-of-30-students-each-student-has-opted-for-at-least-one-of\/","url":"https:\/\/exam.pscnotes.com\/mcq\/in-a-group-of-30-students-each-student-has-opted-for-at-least-one-of\/","name":"In a group of 30 students, each student has opted for at least one of","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:42:16+00:00","dateModified":"2025-06-01T11:42:16+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"Let H be the set of students who opted for Hindi and E be the set of students who opted for English. We are given the total number of students, $|H \\cup E| = 30$ (since each student opted for at least one subject). We are given $|H| = 12$ and $|E| = 22$. The number of students who opted for both subjects is the intersection of H and E, denoted by $|H \\cap E|$. Using the principle of inclusion-exclusion for two sets, we have $|H \\cup E| = |H| + |E| - |H \\cap E|$. Substituting the given values, $30 = 12 + 22 - |H \\cap E|$. This gives $30 = 34 - |H \\cap E|$, so $|H \\cap E| = 34 - 30 = 4$. The number of students who opted for only English is the number of students in E minus the number of students in both H and E. 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