// Numbas version: finer_feedback_settings
{"name": "Binomial Expansion 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "c3", "d1", "d2", "d3", "e1", "e2", "e3", "f1", "f2", "f3", "g1", "g2", "g3", "a3", "b3", "d4", "d5", "e4", "e5", "e6", "e7"], "name": "Binomial Expansion 6", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>In these questions we need to think carefully about which terms in the expansions will contribute to the coefficient that we are trying to find.</p>\n<p>For example, if we have $(a+bx)(c+dx)^n$ and we are looking for the coefficient of the $x^i$ term then the answer will be $a*\\simplify{comb(n,i)}*d^i + b*\\simplify{comb(n, i-1)}*d^{i-1}$</p>\n<p>This makes use of the binomial theorem:</p>\n<p>&nbsp;<img src=\"http://www.purplemath.com/modules/prob/binom01.gif\" width=\"182\" height=\"50\" border=\"0\" alt=\"(a + b)^n = sum[k=0,n][(n over k)a^(n-k)b^k]\"/></p>", "rulesets": {}, "parts": [{"integerPartialCredit": 0, "prompt": "<p>$(1+\\var{a3}x)(1+x)^{\\var{a1}}$ &nbsp; &nbsp; $[x^{\\var{a2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "comb({a1},{a2})+({a3}*comb({a1},{a2}-1))", "minValue": "comb({a1},{a2})+({a3}*comb({a1},{a2}-1))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>$(1-\\var{b3}x)(1+x)^{\\var{b1}}$ &nbsp; &nbsp; $[x^{\\var{b2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "comb({b1},{b2})-(b3*comb({b1},{b2}-1))", "minValue": "comb({b1},{b2})-(b3*comb({b1},{b2}-1))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>$(\\var{c3}-x)(1+x)^{\\var{c1}}$ &nbsp; &nbsp; $[x^{\\var{c2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "(comb({c1},{c2})*{c3})-(comb({c1},{c2}-1))", "minValue": "(comb({c1},{c2})*{c3})-(comb({c1},{c2}-1))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>$(\\var{d4}-\\var{d3}x)(1+\\var{d5}x)^{\\var{d1}}$ &nbsp; &nbsp; $[x^{\\var{d2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "(comb({d1},{d2}))*({d5}^{d2})*({d4})-((comb({d1},{d2}-1)*{d3}*{d5}^({d2}-1)))", "minValue": "(comb({d1},{d2}))*({d5}^{d2})*({d4})-((comb({d1},{d2}-1)*{d3}*{d5}^({d2}-1)))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>$(\\var{e4}+\\simplify{x^{e3}})(\\var{e5} + \\var{e6}x)^{\\var{e1}}$ &nbsp; &nbsp; $[x^{\\var{e2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "({e5}^{e2})({e4}*comb({e1},{e2})*({e7}^{e2})+(comb({e1},{e2}-{e3})*({e7}^({e2}-{e3}))))", "minValue": "({e5}^{e2})({e4}*comb({e1},{e2})*({e7}^{e2})+(comb({e1},{e2}-{e3})*({e7}^({e2}-{e3}))))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>$(1+2x-x^2)(\\var{f3}+x)^{\\var{f1}}$ &nbsp; &nbsp; $[x^{\\var{f2}}]$</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "({f3}^{f1})(comb({f1},{f2})*{f3}^(-{f2}) + 2comb({f1}, {f2}-1)*{f3}^(-{f2}+1)-comb({f1}, {f2}-2)*{f3}^(-({f2}-2)))", "minValue": "({f3}^{f1})(comb({f1},{f2})*{f3}^(-{f2}) + 2comb({f1}, {f2}-1)*{f3}^(-{f2}+1)-comb({f1}, {f2}-2)*{f3}^(-({f2}-2)))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Find the coefficient of the term indicated in the square brackets in the expansion of each of the following expressions below</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "random(4..9 except c1 except b1 except a1 except d1 except e1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "floor(f1/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "e5": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e5", "description": ""}, "b1": {"definition": "random(2..9 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "floor(2b1/3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "random(2..5 except a3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d4": {"definition": "random(2..6 except d3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d4", "description": ""}, "d5": {"definition": "random(2..5 except d3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d5", "description": ""}, "d2": {"definition": "floor(2d1/3+1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "random(2..5 except c3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d1": {"definition": "random(2..9 except c1 except b1 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "g3": {"definition": "random(2..5 except 3)", "templateType": "anything", "group": "Ungrouped variables", "name": "g3", "description": ""}, "g2": {"definition": "floor(2g1/3)", "templateType": "anything", "group": "Ungrouped variables", "name": "g2", "description": ""}, "g1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}, "a1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "floor(a1/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "c3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "e4": {"definition": "random(2..5 except e3)", "templateType": "anything", "group": "Ungrouped variables", "name": "e4", "description": ""}, "c1": {"definition": "random(2..9 except b1 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e6": {"definition": "random(2..5 except e5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e6", "description": ""}, "e1": {"definition": "random(3..9 except c1 except b1 except a1 except d1)", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "floor(2e2/3)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "e2": {"definition": "floor(2e1/3)", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}, "e7": {"definition": "e6/e5", "templateType": "anything", "group": "Ungrouped variables", "name": "e7", "description": ""}, "c2": {"definition": "floor(c1/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}