// Numbas version: exam_results_page_options
{"name": "Counting 1: string enumeration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Counting 1: string enumeration", "tags": [], "metadata": {"description": "<p>Simple counting exercise, with combinations</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>A randomised string of $\\var{n}$ digits is made using only the non-zero digits $1,2,3,4,5,6,7,8,9$.</p>\n<p>When each digit is chosen independently, the number of ways to choose the string is just the product of the number of ways to choose each of the digits.</p>\n<p style=\"text-align: center;\"># of ways to choose string = $\\displaystyle \\prod\\limits_{i=1}^{\\var{n}}$ (# ways to choose $i$th digit).</p>", "advice": "<p>(a) # of ways to choose string $=$ $\\displaystyle \\prod\\limits_{i=1}^{\\var{n}}$ (# ways to choose $i$th digit)&nbsp;$=$ $\\displaystyle \\prod\\limits_{i=1}^{\\var{n}} 9 =&nbsp; {9^\\var{n} }$.</p>\n<p>(b) # of ways to choose string&nbsp;$=$ (# of ways of choosing an even digit) $\\times$ (# of ways of choosing an odd digit) $\\times \\displaystyle \\prod\\limits_{i=3}^{\\var{n}}$ (# ways to choose a digit less than $\\var{b}$)&nbsp;$=$ $4 \\times 5 \\times {\\var{b-1}^\\var{n-2}}$.</p>\n<p>(c) # of ways to choose string&nbsp;$=$ (# of ways of placing the 9s) $\\times$ (# of ways of placing the remaining digits) $=$ ${C(\\var{n},\\var{c})}\\times 8^\\var{n-c}$.</p>", "rulesets": {}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(4..8)", "description": "", "templateType": "anything"}, "ex2": {"name": "ex2", "group": "Ungrouped variables", "definition": "sum(map(random(1..9)*10^k,k,0..(n-1)))", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..(n-1))", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(4..8)", "description": "", "templateType": "anything"}, "ex1": {"name": "ex1", "group": "Ungrouped variables", "definition": "sum(map(random(1..9)*10^k,k,0..(n-1)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "not(ex1 = ex2)", "maxRuns": 100}, "ungrouped_variables": ["ex1", "ex2", "n", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Assuming there are no other restrictions, how many strings are possible?</p>", "stepsPenalty": 0, "steps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>It might be helpful to first consider some examples. The chosen string could be</p>\n<p style=\"text-align: center;\">$\\var{ex1}$ or $\\var{ex2}$.</p>\n<p style=\"text-align: left;\">How many ways can you choose the first digit?</p>", "minValue": "9", "maxValue": "9", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many ways can you choose the $\\var{n}$th digit?</p>", "minValue": "9", "maxValue": "9", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "answer": "{9^n}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many strings are possible if the first digit must be even, the second digit odd, and the remaining $\\var{n-2}$ digits are less than $\\var{b}$?</p>", "stepsPenalty": 0, "steps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many ways can an even number be chosen for the first digit?</p>", "minValue": "4", "maxValue": "4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many ways can an odd number be chosen for the second digit?</p>", "minValue": "5", "maxValue": "5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many ways can the remaining $\\var{n-2}$ digits be chosen, given that they must each be less than $\\var{b}$?</p>", "answer": "{(b-1)^(n-2)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "answer": "{(b-1)^(n-2)*4*5}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>How many strings are possible which contain exactly $\\var{c}$ nines?</p>", "stepsPenalty": 0, "steps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Note that each digit is no longer independently chosen. But we can split the question into two independent parts.</p>\n<p>First, count the number of ways to choose positions for each of the $\\var{c}$ nines. Since the <strong>order of the positions does not matter</strong> (all the nines look identical), this is a combination. Remember you can use the NUMBAS command <code>comb(n,r)</code> for $C(n,r)$.<code></code></p>", "answer": "{comb(n,c)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Then, count the number of ways to choose the digits for the rest of the string. Remember, you can't use the digit nine anymore, so there are 8 choices for each of the remaining&nbsp;unfilled positions.&nbsp;</p>", "answer": "{8^(n-c)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "answer": "{comb(n,c)*8^(n-c)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}