document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=360325>Question 360325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a random arrangement of letters of word VIOLENT how many ways are there that the vowels occupy odd positions only </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #257124 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=257124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> without any restrictions, the number of possible ways to arrange the letters VIOLENT would be 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5040\r<BR>\n" );
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document.write( "this can only happen in one way.\r<BR>\n" );
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document.write( "this assumes that each letter can only be used once.\r<BR>\n" );
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document.write( "since there are 3 vowels and 4 consonants, and the vowels can only be in odd positions, then the possible ways this can happen are provided by the following formula:\r<BR>\n" );
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document.write( "number of possible ways to get sets of 3 out of a total of 4 is:\r<BR>\n" );
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document.write( "4! / (3! * 1!) = (4*3!)/3! = 4\r<BR>\n" );
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document.write( "the general equation for this is:\r<BR>\n" );
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document.write( "n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "n = 4<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "(n-x) = 1\r<BR>\n" );
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document.write( "the number of combinations in each one of these ways is equal to 4! * 3! = 144\r<BR>\n" );
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document.write( "since there are 4 ways that can happen as we just calculated above, then the total ways you can get 3 vowels in only the odd position is 4 * 144 = 576.\r<BR>\n" );
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document.write( "the 4 possible positions the vowels can be in are shown as follows:\r<BR>\n" );
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document.write( "x shows the position of the vowel.<BR>\n" );
document.write( "c shows the position of the consonant.\r<BR>\n" );
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document.write( "position          1   2   3   4   5   6   7\r\n" );
document.write( "way number 1      x   c   x   c   x   c   c\r\n" );
document.write( "way number 2      x   c   x   c   c   c   x\r\n" );
document.write( "way number 3      x   c   c   c   x   c   x\r\n" );
document.write( "way number 4      c   c   x   c   x   c   x\r\n" );
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document.write( "the number of possible combinations in each of the ways is calculated as follows:\r<BR>\n" );
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document.write( "we'll take way number 1 to show you. \r<BR>\n" );
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document.write( "the other ways work the same.\r<BR>\n" );
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document.write( "way number 1\r<BR>\n" );
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document.write( "vowel in 1st and 3d and 5th position.\r<BR>\n" );
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document.write( "you have 3 vowels.\r<BR>\n" );
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document.write( "the first vowel position can take any 3 of them.<BR>\n" );
document.write( "the second vowel position can take any 2 of them.<BR>\n" );
document.write( "the third vowel position can take any 1 of them.\r<BR>\n" );
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document.write( "total possible combinations are therefore 3*2*1 = 6 = 3! for the vowels.\r<BR>\n" );
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document.write( "you have 4 consonants.\r<BR>\n" );
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document.write( "the first consonant position can take any 4 of them.<BR>\n" );
document.write( "the second consonant position can take any 3 of them.<BR>\n" );
document.write( "the third consonant position can take any 2 of them.<BR>\n" );
document.write( "the fourth consonant position can take any 1 of them.\r<BR>\n" );
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document.write( "total possible combinations are therefore 4*3*2*1 = 24  = 4! for the consonants.\r<BR>\n" );
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document.write( "for each possible combination of vowels, there are 24 possible combinations of consonants.\r<BR>\n" );
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document.write( "since there are 6 possible combinations of vowels, then the total number of possible combinations of vowels and consonants for way number 1 is 6 * 24 = 144.\r<BR>\n" );
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document.write( "4 possible ways means 4 * 144 = 576.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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