// Numbas version: exam_results_page_options
{"name": "Full binomial expansion", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "tags": [], "ungrouped_variables": ["a", "b", "n", "c", "k", "ellipsis"], "variable_groups": [], "advice": "<p>The binomial series expansion for an expression of the form \\((a+bx)^n\\) where \\(n\\) is a Natural number is given by:</p>\n<p style=\"padding-left: 30px;\">\\((a+bx)^n=\\tbinom{n}{0}(a)^n(bx)^{0}+\\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...+\\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...+\\tbinom{n}{n}(a)^{0}(bx)^{n}\\)</p>\n<p>In this example &nbsp;\\(n=\\var{n}\\), &nbsp;\\(a=\\var{a}\\) &nbsp;and &nbsp;\\(b=\\var{b}\\).</p>\n<p>So the binomial series expansion is:</p>\n<p style=\"padding-left: 30px;\">&nbsp;\\(\\var{a}^{\\var{n}}+\\tbinom{\\var{n}}{\\var{1}}\\times\\var{a}^{\\var{n}-1}\\times(\\var{b}x)^{1}+\\tbinom{\\var{n}}{2}\\times\\var{a}^{\\var{n}-2}\\times(\\var{b}x)^{2}+\\var{ellipsis}\\tbinom{\\var{n}}{\\var{n}}\\times\\var{a}^{\\var{n}-\\var{n}}\\times(\\var{b}x)^\\var{n}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}^{n}+{n}*{a}^({n}-1)*{b}x+{n}*{n-1}*{a}^{{n}-2}*{{b}^2}/2x^2+{n}*{n-1}*{n-2}*{a}^{{n}-3}*{{b}^3}/6x^3+{n}*{n-1}*{n-2}*{n-3}*{a}^{{n}-4}*{{b}^4}/24x^4+{n}*{n-1}*{n-2}*{n-3}*{n-4}*{a}^{{n}-5}*{{b}^5}/120x^5}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "metadata": {"description": "<p>Find the binomial expansion of the following:</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "rulesets": {}, "variables": {"c": {"description": "", "definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}", "group": "Ungrouped variables", "name": "c", "templateType": "anything"}, "b": {"description": "", "definition": "random(-4..-1 except 0)", "group": "Ungrouped variables", "name": "b", "templateType": "anything"}, "n": {"description": "", "definition": "random(3..5#1)", "group": "Ungrouped variables", "name": "n", "templateType": "randrange"}, "k": {"description": "", "definition": "random(2..{n}-1)", "group": "Ungrouped variables", "name": "k", "templateType": "anything"}, "a": {"description": "", "definition": "random(2..6#1)", "group": "Ungrouped variables", "name": "a", "templateType": "randrange"}, "ellipsis": {"description": "", "definition": "if(n>3,\" ... + \",\"\")", "group": "Ungrouped variables", "name": "ellipsis", "templateType": "anything"}}, "parts": [{"checkingType": "absdiff", "prompt": "<p>Write out the&nbsp;full binomial expansion:</p>", "notallowed": {"showStrings": false, "strings": ["(", ")"], "message": "", "partialCredit": 0}, "failureRate": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showPreview": false, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "expectedVariableNames": [], "answerSimplification": "all, canonicalOrder, !noLeadingMinus", "scripts": {}, "unitTests": [], "musthave": {"showStrings": false, "strings": ["x^2", "x^3"], "message": "", "partialCredit": 0}, "answer": "{a}^{n}+{n}*{a}^({n}-1)*{b}*x+({n}*{n-1}*{a}^({n}-2)*{{b}^2}/2)*x^2+{n}*{n-1}*{n-2}*{a}^({n}-3)*{{b}^3}/6*x^3+{n}*{n-1}*{n-2}*{n-3}*{a}^({n}-4)*{{b}^4}/24*x^4++{n}*{n-1}*{n-2}*{n-3}*{n-4}*{a}^({n}-5)*{{b}^5}/120*x^5", "marks": "5", "vsetRange": [0, 1], "checkVariableNames": false, "type": "jme"}], "statement": "<p>Given the expression $\\simplify{({a}+{b}x)^{n}}$</p>", "name": "Full binomial expansion", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}