document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151170>Question 1151170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2. If the permutation of the word WHITE is selected at random, how many of the<BR>\n" );
document.write( "permutations<BR>\n" );
document.write( "i. Begins with a consonant?<BR>\n" );
document.write( "ii. Ends with a vowel?<BR>\n" );
document.write( "iii. Has a consonant and vowels alternating? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772819 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=772819\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "i.<BR>\n" );
document.write( "Consonants to choose from = {W, H, T}<BR>\n" );
document.write( "Vowels to choose from = {I, E}\r<BR>\n" );
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document.write( "We have 3 consonants to fill slot 1. <BR>\n" );
document.write( "Then we have 4 choices for slot 2<BR>\n" );
document.write( "Then 3 choices for slot 3<BR>\n" );
document.write( "2 choices for slot 4<BR>\n" );
document.write( "1 choice for slot 5\r<BR>\n" );
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document.write( "Overall, 3*4*3*2*1 = 3*4! = 3*24 = 72\r<BR>\n" );
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document.write( "There are 72 permutations that begin with a consonant.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>72</font>\r<BR>\n" );
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document.write( "ii.\r<BR>\n" );
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document.write( "Slot 5 has 2 choices, either an E or an I.<BR>\n" );
document.write( "After slot 5 is filled, we have 4 choices for slot 1<BR>\n" );
document.write( "Then 3 choices for slot 2<BR>\n" );
document.write( "2 choices for slot 3<BR>\n" );
document.write( "1 choice for slot 4\r<BR>\n" );
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document.write( "2*4*3*2*1 = 2*4! = 2*24 = 48\r<BR>\n" );
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document.write( "There are 48 permutations that have a vowel as the last letter.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>48</font>\r<BR>\n" );
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document.write( "iii.\r<BR>\n" );
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document.write( "define two variable placeholders<BR>\n" );
document.write( "c = consonant from the list {W, H, T}<BR>\n" );
document.write( "V = vowel from the list {I, E}\r<BR>\n" );
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document.write( "We want the consonants c and the vowels V to alternate. If we start with a consonant, then we must have a vowel next, and then another consonant, etc etc.\r<BR>\n" );
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document.write( "That pattern would be <BR>\n" );
document.write( "cVcVc\r<BR>\n" );
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document.write( "We can't start with a vowel because we'd have<BR>\n" );
document.write( "VcVcc<BR>\n" );
document.write( "we run out of vowels once we reach slot 3.\r<BR>\n" );
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document.write( "And of course we can't have something like this<BR>\n" );
document.write( "ccVcV<BR>\n" );
document.write( "because of the adjacent consonants in slots 1 and 2.\r<BR>\n" );
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document.write( "So again we only have something of the form<BR>\n" );
document.write( "cVcVc\r<BR>\n" );
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document.write( "Consider just the consonants for now. We basically have 3 slots to fill and 3 items to fill them. There are 3! = 3*2*1 = 6 ways to do this. Order matters.\r<BR>\n" );
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document.write( "Then we have 2 other slots for the vowels, so 2! = 2*1 = 2 ways to fill the vowel slots. \r<BR>\n" );
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document.write( "Overall, there are 6*2 = 12 ways to have a permutation of letters such that the consonants and vowels alternate.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>12</font><BR>\n" );
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document.write( " </SPAN>\n" );
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