Algebra.Com's Answer #852244 by ikleyn(52790)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A scholarship committee has $3000 to award this year and has 12 qualified candidates.
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document.write( "Fran thinks that individual awards of $1500, $1000, and $500 would be appropriate,
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document.write( "while Pat thinks that 3 awards of $1000 each would seem logical. Answer the following questions.
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document.write( "a) Count the number of possible outcomes for each plan.
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document.write( "b) Explain which plan would allow the scholarship committee to reach a consensus most quickly.
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document.write( "If we consider Fran's plan, then the number of all different possible and distinguishable triples\r\n" );
document.write( "is the product 12*11*10 = 1320 of three integer numbers starting from 12 in the descending order.\r\n" );
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document.write( "It is so, because the order inside each such triple does matter:\r\n" );
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document.write( "for example, triple (Alex, Berta, Carl) is different from triple (Berta, Alex, Carl),\r\n" );
document.write( "because the participants obtain different awards.\r\n" );
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document.write( "If we consider Pat's plan, then the number of all different possible and distinguishable triples\r\n" );
document.write( "is the number of combinations  =  =  = 220 of three participants\r\n" );
document.write( "taken at a time from 12 participants.\r\n" );
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document.write( "It is so, because the order inside each such triple does NOT matter:\r\n" );
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document.write( "for example, triple (Alex, Berta, Carl) is not distinguishable from triple (Berta, Alex, Carl),\r\n" );
document.write( "because the participants obtain the same awards.\r\n" );
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document.write( "Thus part (a) is solved completely and question (a) is answered, in full.\r\n" );
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document.write( "It is a good question and a good problem, because it teaches you \r\n" );
document.write( "to distinct between permutations and combinations, depending on context.\r\n" );
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document.write( " Regarding question (b), I do not understand its meaning.\r\n" );
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document.write( "May be, the most fair/equitable way is to distribute all $3000 equally between all 12 participants ?\r\n" );
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document.write( "May be, the most fair way is to award with $3000 the most qualified candidate ?\r\n" );
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document.write( "I don't know which way is better.\r\n" );
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document.write( "Flip a coin. Or declare a war - on your choice.\r\n" );
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document.write( "To speak seriously, the way how this question (b) is posed in the post is not a mathematical.\r\n" );
document.write( "In my view, it would be much better to exclude question (b) from the problem, at all.\r\n" );
document.write( "Why ? - - - Because there is no room for nonsense in Math problems.\r\n" );
document.write( "If you remove this question from the post, then the problem will become ideal from the educative point of view.\r\n" );
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