// Numbas version: finer_feedback_settings
{"name": "Teilmengen (Kinder)", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Teilmengen (Kinder)", "tags": [], "metadata": {"description": "<p>Verschiedene Fragen zu Anzahlen von Teilmengen.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Es sei $K$ eine Menge von&nbsp;$\\var{sum}$ Kindern, $M$ die Teilmenge der M&auml;dchen mit $|M|=\\var{m}$, $B$ die Teilmenge der Burschen mit $|B|=\\var{b}$.</p>", "advice": "<p>(a) F&uuml;r diese Frage ist es egal, ob es Burschen oder M&auml;dchen sind. Es sind also aus $n=\\var{sum}$ Elementen $k=\\var{(sum+1)/2}$ ohne Wiederholung und ohne Beachtung der Reihenfolge, d.h. Kombinationen (ohne Wiederholung), also $\\binom{\\var{sum}}{\\var{(sum+1)/2}}=\\var{a_result}$.</p>\n<p></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 679px; width: 0px; top: 29px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 664px; width: 50px; top: 14px; height: 20px; font-size: 10px; color: black;\"></span></p>\n<p>(b) F&uuml;r diese Frage sind die Teilgruppen \"M&auml;dchen\" und \"Burschen\" getrennt zu betrachten, sonst gilt daselbe wie bei (a): Innerhalb der Gruppen erfolgt eine Auswahl ohne Wiederholung und ohne Beachtung der Reihenfolge, man erh&auml;lt also&nbsp;$\\binom{\\var{m}}{\\var{mini}}$ f&uuml;r die&nbsp;M&auml;dchen und&nbsp;&nbsp;$\\binom{\\var{b}}{\\var{mini}}$ f&uuml;r die Burschen, diese beiden Zahlen sind zu multiplizieren (Entscheidungen sind keine Alternativen und h&auml;ngen nicht voneinander ab), was $\\var{b_result}$ ergibt.</p>\n<p>(c) Es gibt 1 M&ouml;glichkeit, alle M&auml;dchen auszuw&auml;hlen, diese ist dann noch mit den insgesamt (alle Teilmengen) $2^{\\var{b}}=\\var{c_result}$ M&ouml;glichkeiten zu multiplizieren, $0,1,2,\\ldots \\var{b}$ Burschen auszuw&auml;hlen.</p>\n<p>(d) Genau die H&auml;lfte aller Teilmengen enth&auml;lt weniger als $\\var{(sum+1)/2}$, genau die H&auml;lfte mindestens $\\var{(sum+1)/2}$ Kinder. Die Anzahl aller Teilmengen einer $\\var{sum}$-elementigen Menge ist $2^{\\var{sum}}$, die H&auml;lfte davon ist&nbsp;$2^{\\var{sum-1}}=\\var{d_result}$.</p>", "rulesets": {}, "extensions": ["stats"], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "sum-b ", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(3..6)", "description": "", "templateType": "anything"}, "b_result": {"name": "b_result", "group": "Ungrouped variables", "definition": "combination(m,mini)*combination(b,mini)", "description": "", "templateType": "anything"}, "a_result": {"name": "a_result", "group": "Ungrouped variables", "definition": "combination(sum,(sum+1)/2)", "description": "", "templateType": "anything"}, "c_result": {"name": "c_result", "group": "Ungrouped variables", "definition": "2^b", "description": "", "templateType": "anything"}, "d_result": {"name": "d_result", "group": "Ungrouped variables", "definition": "2^(sum-1)", "description": "", "templateType": "anything"}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "2*random(5..7)+  1", "description": "", "templateType": "anything"}, "half": {"name": "half", "group": "Ungrouped variables", "definition": " (sum+1)/2  ", "description": "", "templateType": "anything"}, "mini": {"name": "mini", "group": "Unnamed group", "definition": "min(b,m)-1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "m", "b_result", "a_result", "c_result", "d_result", "sum", "half"], "variable_groups": [{"name": "Unnamed group", "variables": ["mini"]}], "functions": {}, "preamble": {"js": "", "css": ".copyright-footer{\n  display:none;}"}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Wie viele Teilmengen von $K$ enthalten genau $\\var{(sum+1)/2}$ Kinder?&nbsp;</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 470px; width: 0px; top: 30px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 455px; width: 50px; top: 15px; height: 20px; font-size: 10px; color: black;\"></span></p>", "minValue": "a_result", "maxValue": "a_result", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Wie viele Teilmengen von $K$ enthalten genau $\\var{mini}$ M&auml;dchen und genau $\\var{mini}$ Burschen?</p>", "minValue": "b_result", "maxValue": "b_result", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Wie viele Teilmengen von $K$ enthalten&nbsp;alle&nbsp;$\\var{m}$ M&auml;dchen?&nbsp;</p>", "minValue": "c_result", "maxValue": "c_result", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Wie viele Teilmengen von $K$ enthalten mindestens&nbsp;$\\var{(sum+1)/2}$ Kinder?&nbsp;</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 39px; width: 0px; top: 34px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 24px; width: 50px; top: 19px; height: 20px; font-size: 10px; color: black;\"></span></p>", "minValue": "d_result", "maxValue": "d_result", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}