// Numbas version: exam_results_page_options
{"name": "Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "tags": [], "statement": "<p>Given the expression \\((\\var{a}+\\var{b}x)^{\\var{n}}\\)</p>", "parts": [{"prompt": "<p>By using the binomial series expansion, calculate the coefficient of \\(x^{\\var{k}}\\) &nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "allowFractions": true, "showFeedbackIcon": true, "type": "numberentry", "marks": 1, "variableReplacements": [], "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "maxValue": "{c}", "mustBeReducedPC": 0, "minValue": "{c}", "scripts": {}}], "variableReplacements": [], "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "scripts": {}}], "variable_groups": [], "name": "Binomial series for Natural exponent", "variables": {"k": {"definition": "random(2..{n}-1)", "description": "", "name": "k", "templateType": "anything", "group": "Ungrouped variables"}, "b": {"definition": "random(2..12#1)", "description": "", "name": "b", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"definition": "random(2..10#1)", "description": "", "name": "a", "templateType": "randrange", "group": "Ungrouped variables"}, "c": {"definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}", "description": "", "name": "c", "templateType": "anything", "group": "Ungrouped variables"}, "n": {"definition": "random(4..9#1)", "description": "", "name": "n", "templateType": "randrange", "group": "Ungrouped variables"}}, "rulesets": {}, "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;\\(k=\\var{k}\\), &nbsp;\\(a=\\var{a}\\) &nbsp;and &nbsp;\\(b=\\var{b}\\).</p>\n<p>So the coefficient of \\(x^{\\var{k}}\\) is given by \\(\\tbinom{\\var{n}}{\\var{k}}*\\var{a}^{\\var{n}-\\var{k}}*\\var{b}^{\\var{k}}=\\var{c}\\).</p>\n<p style=\"padding-left: 30px;\"></p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Binomial series for Natural exponent</p>"}, "ungrouped_variables": ["a", "b", "n", "c", "k"], "functions": {}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}