document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=360323>Question 360323</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the letters of the word \" REGULATIONS \" be arranged at random, what is the chance that there will be exactly 4 letters between the R and the E </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #257132 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=257132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the word is REGULATIONS\r<BR>\n" );
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document.write( "since there are 11 characters in the word, then those characters can be arranged in 11! = 39916800 ways randomly.\r<BR>\n" );
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document.write( "the number of ways that you can get 4 letters between any 2 specified letters are:\r<BR>\n" );
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document.write( "?xxxx?xxxxx<BR>\n" );
document.write( "x?xxxx?xxxx<BR>\n" );
document.write( "xx?xxxx?xxx<BR>\n" );
document.write( "xxx?xxxx?xx<BR>\n" );
document.write( "xxxx?xxxx?x<BR>\n" );
document.write( "xxxxx?xxxx?\r<BR>\n" );
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document.write( "? represents the letters that have to have 4 letters between them.<BR>\n" );
document.write( "x represents the other letters.\r<BR>\n" );
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document.write( "that's a total of 6 ways.\r<BR>\n" );
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document.write( "if you wanted to solve this part of problem using a formula, you would need to consider the number of ways as follows:\r<BR>\n" );
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document.write( "2 letters with 4 letters between them equals 1 block of 6 letters that have to be considered as 1 special letter because they are always together.\r<BR>\n" );
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document.write( "you start with 11 letters.  you have 1 block of 6 letters that needs to be considered as 1.\r<BR>\n" );
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document.write( "subtract 6 from 11 and you get 5 remaining letters.\r<BR>\n" );
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document.write( "your letters are B x x x x x\r<BR>\n" );
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document.write( "The B represents the block of 6 letters.\r<BR>\n" );
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document.write( "The total possible ways this can happen is 6! / (1! * 5!) = 6! / 5! = 6.\r<BR>\n" );
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document.write( "Those possible ways are:\r<BR>\n" );
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document.write( "B x x x x <BR>\n" );
document.write( "x B x x x<BR>\n" );
document.write( "x x B x x <BR>\n" );
document.write( "x x x B x<BR>\n" );
document.write( "x x x x B\r<BR>\n" );
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document.write( "if you look at the 6 ways that I showed you and you replace ?xxxx? with B, then you will see that the B represents ?xxxx?.\r<BR>\n" );
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document.write( "So the number of ways that the pattern can present itself is 6.\r<BR>\n" );
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document.write( "the number of possible combinations in each one of those ways is:\r<BR>\n" );
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document.write( "9! * 2! = 725760.\r<BR>\n" );
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document.write( "the number of possible combinatons in all 6 of those ways is therefore:\r<BR>\n" );
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document.write( "6 * 9! * 2! = 6 * 725760 = 4354560\r<BR>\n" );
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document.write( "the probability that will happen is therefore equal to 4354560 / 39916800 = .109090909\r<BR>\n" );
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document.write( "hard to see with big numbers so let's try smaller numbers:\r<BR>\n" );
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document.write( "assume the word is MARK.\r<BR>\n" );
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document.write( "what is the probability that the A and the K will always have 1 letter between them.\r<BR>\n" );
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document.write( "total possible combinations in the word MARK equals 4! = 24.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "MARK<BR>\n" );
document.write( "MAKR<BR>\n" );
document.write( "MRAK<BR>\n" );
document.write( "MRKA<BR>\n" );
document.write( "MKAR<BR>\n" );
document.write( "MKRA\r<BR>\n" );
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document.write( "ARKM<BR>\n" );
document.write( "ARMK<BR>\n" );
document.write( "AKRM<BR>\n" );
document.write( "AKMR<BR>\n" );
document.write( "AMRK<BR>\n" );
document.write( "AMKR\r<BR>\n" );
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document.write( "RMAK<BR>\n" );
document.write( "RMKA<BR>\n" );
document.write( "RAKM<BR>\n" );
document.write( "RAMK<BR>\n" );
document.write( "RKAM<BR>\n" );
document.write( "RKMA\r<BR>\n" );
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document.write( "KMAR<BR>\n" );
document.write( "KMRA<BR>\n" );
document.write( "KAMR<BR>\n" );
document.write( "KARM<BR>\n" );
document.write( "KRMA<BR>\n" );
document.write( "KRAM\r<BR>\n" );
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document.write( "now restrict the letters so there is exactly 1 letter betweeen A and K.\r<BR>\n" );
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document.write( "this can happen in 2 ways.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "?x?x<BR>\n" );
document.write( "x?x?\r<BR>\n" );
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document.write( "? represents either A or K<BR>\n" );
document.write( "x represents either of the remaining letters.\r<BR>\n" );
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document.write( "total possible combinations are therefore equal to 2 * 2! * 2! = 8.\r<BR>\n" );
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document.write( "those possible combinations are:\r<BR>\n" );
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document.write( "A and K are in the 2d and 4th positions.,\r<BR>\n" );
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document.write( "MARK<BR>\n" );
document.write( "MKRA<BR>\n" );
document.write( "RAMK<BR>\n" );
document.write( "RKMA\r<BR>\n" );
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document.write( "A and K are in the 1sd and 3d positions.\r<BR>\n" );
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document.write( "AMKR<BR>\n" );
document.write( "KMAR<BR>\n" );
document.write( "ARKM<BR>\n" );
document.write( "KRAM\r<BR>\n" );
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document.write( "total number of possible ways is 4! = 24<BR>\n" );
document.write( "total number of ways A and K can have exactly one letter between them is 2*2!*2! = 8.\r<BR>\n" );
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document.write( "probability of that happening is 8 / 24 = 1/3.\r<BR>\n" );
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document.write( "same concept only smaller numbers that can be shown easier.\r<BR>\n" );
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document.write( "` </SPAN>\n" );
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