document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=285534>Question 285534</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Provide an example of a situation that describes a permutation?\r<BR>\n" );
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document.write( "Provide a situation that describes a combination?<BR>\n" );
document.write( "How many different plates are possible in the United Kingdom?<BR>\n" );
document.write( "Identify the probability that a randomly selected plate is AA 51 ZZZ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #206950 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=206950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 4 runners in a race.\r<BR>\n" );
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document.write( "Only 3 will get a prize.\r<BR>\n" );
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document.write( "What are the possible ways the prize can be distributed?\r<BR>\n" );
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document.write( "Since order is important, this will be a permutation.\r<BR>\n" );
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document.write( "You will determine how many ordered sets of 3 can you draw from the 5 runners.\r<BR>\n" );
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document.write( "The formula will be:\r<BR>\n" );
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document.write( "n! / (n-x)! where n is the total number of runners and x is the number of runners in the set that you will be drawing.\r<BR>\n" );
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document.write( "n = 4<BR>\n" );
document.write( "x = 3\r<BR>\n" );
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document.write( "n! / (n-x)! = (4*3*2*1) / (1) = 4*3*2 = 24\r<BR>\n" );
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document.write( "If we give each runner a letter, then the possible letters for the 4 runners are:\r<BR>\n" );
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document.write( "a,b,c,d\r<BR>\n" );
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document.write( "The possible sets of 3 that we can draw are:\r<BR>\n" );
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document.write( "a,b,c *****<BR>\n" );
document.write( "a,c,b<BR>\n" );
document.write( "b,a,c<BR>\n" );
document.write( "b,c,a<BR>\n" );
document.write( "c,a,b<BR>\n" );
document.write( "c,b,a<BR>\n" );
document.write( "a,b,d *****<BR>\n" );
document.write( "a,d,b<BR>\n" );
document.write( "b,a,d<BR>\n" );
document.write( "b,d,a<BR>\n" );
document.write( "d,a,b<BR>\n" );
document.write( "d,b,a<BR>\n" );
document.write( "a,c,d *****<BR>\n" );
document.write( "a,d,c<BR>\n" );
document.write( "c,a,d<BR>\n" );
document.write( "c,d,a<BR>\n" );
document.write( "d,a,c<BR>\n" );
document.write( "d,c,a<BR>\n" );
document.write( "b,c,d *****<BR>\n" );
document.write( "b,d,c<BR>\n" );
document.write( "c,b,d<BR>\n" );
document.write( "c,d,b<BR>\n" );
document.write( "d,b,c<BR>\n" );
document.write( "d,c,b\r<BR>\n" );
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document.write( "What are the possible ways in which runners will be eligible for a prize?\r<BR>\n" );
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document.write( "Since order is not important, this will be a combination.\r<BR>\n" );
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document.write( "You will determine how many unordered sets of 3 can you draw from the 5 runners.\r<BR>\n" );
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document.write( "The formula will be:\r<BR>\n" );
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document.write( "n! / ((n-x)!*x!) where n is the total number of runners and x is the number of runners in the set that you will be drawing.\r<BR>\n" );
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document.write( "n = 4<BR>\n" );
document.write( "x = 3\r<BR>\n" );
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document.write( "n! / ((n-x)!*x!) = 4! / (1!*3!) = (4*3*2*1) / (1*3*2*1) = 4\r<BR>\n" );
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document.write( "The 4 possible ways are:\r<BR>\n" );
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document.write( "a,b,c<BR>\n" );
document.write( "a,b,d<BR>\n" );
document.write( "a,c,d<BR>\n" );
document.write( "b,c,d\r<BR>\n" );
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document.write( "Notice that the 4 possible ways the runners will be eligible for a prize is the asterisked permutations above.\r<BR>\n" );
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document.write( "In other words, if you disregard the order in which they came in, one of the possible combinations is a,b,c.\r<BR>\n" );
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document.write( "If you then wanted to determine the possible ways they could come in, you would then get:\r<BR>\n" );
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document.write( "a,b,c<BR>\n" );
document.write( "a,c,b<BR>\n" );
document.write( "b,a,c<BR>\n" );
document.write( "b,c,a<BR>\n" );
document.write( "c,a,b<BR>\n" );
document.write( "c,b,a\r<BR>\n" );
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document.write( "The same 3 runners could come in 6 different ways.\r<BR>\n" );
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document.write( "The combination got you who the runners were.<BR>\n" );
document.write( "The permutation got you who the runners were and the order in which those same 3 runners could come in.\r<BR>\n" );
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document.write( "If you will notice in the formulas:\r<BR>\n" );
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document.write( "The combination formula was:\r<BR>\n" );
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document.write( "n! / ((n-x)! * x!)\r<BR>\n" );
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document.write( "The permutation formula was:\r<BR>\n" );
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document.write( "n! / (n-x)!\r<BR>\n" );
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document.write( "The x! in the divisor of the combination formula is missing in the permutation formula.\r<BR>\n" );
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document.write( "That x! is the number of ways the same set of runners could finish.\r<BR>\n" );
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document.write( "There were 3 runners and they could finish in 3! ways.\r<BR>\n" );
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document.write( "3! = 3*2*1 = 6 ways the 3 runners could finish.\r<BR>\n" );
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document.write( "The number of combinations was 4! / (1!*3!) = 24 / 6 = 4.\r<BR>\n" );
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document.write( "The number of permutations was 4! / 1! = 24 / 1 = 24.\r<BR>\n" );
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document.write( "Combinations only got the people regardless of how they finished.\r<BR>\n" );
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document.write( "Permutations got the people and how they finished.\r<BR>\n" );
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document.write( "LICENSE PLATES IN GREAT BRITAIN\r<BR>\n" );
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document.write( "The possible letters do not add up to the total letters in the alphabet but I don't think that's what you want.\r<BR>\n" );
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document.write( "See this reference for what I am talking about:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://en.wikipedia.org/wiki/Vehicle_registration_plates_of_the_United_Kingdom#Current_system\" target = \"_blank\">http://en.wikipedia.org/wiki/Vehicle_registration_plates_of_the_United_Kingdom#Current_system</a>\r<BR>\n" );
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document.write( "Assume that all letters are valid, each character can have 26 letters and each number can have 10 numbers.\r<BR>\n" );
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document.write( "The total number of possibilities would be:\r<BR>\n" );
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document.write( "26*26*10*10*26*26*26 = 26^5*10^2 = 11881376700 total possible license plate numbers.\r<BR>\n" );
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document.write( "The probability of getting AA 51 ZZZ would be 1 out of 11881376700 = 8.416533573 * 10^-10\r<BR>\n" );
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document.write( "Looked at another way, the probability would be:\r<BR>\n" );
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document.write( "1/26 * 1/26 * 1/10 * 1/10 * 1/26 * 1/26 * 1/26 = 1/26*26*10*10*26*26*26) = 1/(26^5*10^2) = 1/1188137600 = 8.416533573^(-10).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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