// Numbas version: exam_results_page_options {"name": "Simon's copy of Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Simon's copy of Binomial series for Natural exponent", "statement": "

Given the expression \$$(\\var{a}+\\var{b}x)^{\\var{n}}\$$

", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}", "name": "c", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"definition": "random(2..10#1)", "name": "a", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b": {"definition": "random(2..12#1)", "name": "b", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "n": {"definition": "random(4..9#1)", "name": "n", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k": {"definition": "random(2..{n}-1)", "name": "k", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "rulesets": {}, "ungrouped_variables": ["a", "b", "n", "c", "k"], "preamble": {"css": "", "js": ""}, "tags": [], "functions": {}, "advice": "

The binomial series expansion for an expression of the form \$$(a+bx)^n\$$ where \$$n\$$ is a Natural number is given by:

\n

\$$(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}\$$

\n

In this example  \$$n=\\var{n}\$$, \$$a=\\var{a}\$$  and  \$$b=\\var{b}\$$.

\n

\n

Since we only require the coefficient of \$$x^{\\var{k}}\$$, we just require the term where \$$k=\\var{k}\$$.

\n

So the coefficient of \$$x^{\\var{k}}\$$ is given by \$$\\tbinom{\\var{n}}{\\var{k}}\\times\\var{a}^{\\var{n}-\\var{k}}\\times\\var{b}^{\\var{k}}=\\var{c}\$$.

\n

", "parts": [{"showFeedbackIcon": true, "gaps": [{"maxValue": "{c}", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "minValue": "{c}", "allowFractions": true, "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "type": "numberentry", "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false}], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "prompt": "

By using the binomial series expansion, calculate the coefficient of \$$x^{\\var{k}}\$$  [[0]]

", "sortAnswers": false, "showCorrectAnswer": true, "unitTests": [], "marks": 0, "type": "gapfill", "variableReplacements": [], "scripts": {}}], "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Binomial series for Natural exponent

"}, "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/"}]}