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{"name": "Simon's copy of Number of multisets from a given set,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variable_groups": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"marks": 0, "showCorrectAnswer": true, "customName": "", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "prompt": "<p>Number of ways = [[0]]</p>", "variableReplacements": [], "customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "gaps": [{"marks": 2, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "customName": "", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "type": "numberentry", "unitTests": [], "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "mustBeReducedPC": 0, "variableReplacements": [], "allowFractions": false, "customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "maxValue": "ans", "showFractionHint": true, "minValue": "ans", "showFeedbackIcon": true}], "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Simon's copy of Number of multisets from a given set,", "variables": {"p": {"description": "", "definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "p"}, "b": {"description": "", "definition": "if(tb<=p,p+1,tb)", "templateType": "anything", "group": "Ungrouped variables", "name": "b"}, "ans": {"description": "", "definition": "comb(p+b-1,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans"}, "tb": {"description": "", "definition": "random(7..15)", "templateType": "anything", "group": "Ungrouped variables", "name": "tb"}}, "preamble": {"css": "", "js": ""}, "advice": "<p>We are not interested in the order in which the cars were sold, but in the number of cars each salesperson sold.</p>\n<p></p>\n<p>One way to think about this is the 'stars and bars' principle. Suppose we had 4 salesmen and 7 cars. Then we could represent the 7 cars using 7 stars in a row:</p>\n<p>$* * * * * **$</p>\n<p>We could represent the split in sales between the 4 salesmen using 4-1 = 3 bars.</p>\n<p>e.g.</p>\n<p>$* |* *| * * *|*$ represents the first salesman selling 1 car, the second selling 2, the third selling 3 and the fourth selling 1</p>\n<p>$* ||* * * * **|$ represents the first salesman selling 1 car, the second selling 0, the third selling 6 and the fourth selling 0</p>\n<p></p>\n<p>Note we have 7+4-1 = 10 stars and bars in total</p>\n<p>Then the total number of ways of 4 salesmen selling 7 cars is the same as the number of ways of choosing 7 items to be stars out of the 10 items in total.</p>\n<p>i.e. $ {7+4-1 \\choose 7}= {10 \\choose 7} = 120$</p>\n<p></p>\n<p></p>\n<p>Similarly for our example, the number of ways&nbsp;of $\\var{p}$&nbsp;salesmen selling $\\var{b}$ cars is given by:</p>\n<p></p>\n<p>\\[\\binom{\\var{b}+\\var{p}-1}{\\var{b}}=\\binom{\\var{p+b-1}}{\\var{b}}=\\var{ans}\\]</p>", "ungrouped_variables": ["p", "b", "tb", "ans"], "rulesets": {}, "statement": "<p>A car dealership employs $\\var{p}$ salespeople. A salesperson receives a &pound;100 bonus for each car they sell.</p>\n<p>Yesterday, the dealership sold $\\var{b}$ cars.</p>\n<p>In how many ways could this happen?</p>\n<p>(Consider two scenarios different if they result in different bonus payments).</p>", "tags": [], "extensions": [], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}