document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=390902>Question 390902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many ways are there to rearrange the letters in the word  aptitude, if the first and last letter must each be a vowel? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #277240 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=277240\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'll take a stab at this:<BR>\n" );
document.write( "A list of the possible 1st letters with the possible last letters:<BR>\n" );
document.write( "a - - - -i, u, or e (3 choices)<BR>\n" );
document.write( "i - - - - a, u, or e (3 choices)<BR>\n" );
document.write( "u - - - -a, i, or e (3 choices)<BR>\n" );
document.write( "e - - - -a, u, or i (3 choices)<BR>\n" );
document.write( "This is 3 choices times 4, or 12 choices for the 1st and last<BR>\n" );
document.write( "letters being vowels.<BR>\n" );
document.write( "Suppose I pick<BR>\n" );
document.write( "a - - ptitud - - e for my 1st choice<BR>\n" );
document.write( "For the 2nd letter I have 6 choices<BR>\n" );
document.write( "For the 3rd letter I have 5 choices remaining<BR>\n" );
document.write( "For the 4th letter I have 4 choices remaining<BR>\n" );
document.write( "For the 5th letter I have 3 choices remaining<BR>\n" );
document.write( "For the 6th letter I have 2 choices remaining<BR>\n" );
document.write( "For the 7th letter I have 1 choice remaining<BR>\n" );
document.write( "Since the letters are 1st AND 2nd AND 3rd . . .etc.<BR>\n" );
document.write( "I must multiply the choices<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A5%2A4%2A3%2A2%2A1+=+720 ALIGN=MIDDLE  ALT=\"6%2A5%2A4%2A3%2A2%2A1+=+720\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For each of my choices I have treated t and t as different<BR>\n" );
document.write( "letters, but they are not different, so each arrangement is<BR>\n" );
document.write( "counted twice, like:<BR>\n" );
document.write( "a - - tuitdp - - e and<BR>\n" );
document.write( "a - - tuitdp - - e<BR>\n" );
document.write( "So if I divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I'll get the actual number of<BR>\n" );
document.write( "arrangements I want<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720%2F2+=+360 ALIGN=MIDDLE  ALT=\"720%2F2+=+360\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each of these <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360 ALIGN=MIDDLE  ALT=\"360\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be matched with 1 of the 12<BR>\n" );
document.write( "ways I can pick letters for 1st and last, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%2A12+=+4320 ALIGN=MIDDLE  ALT=\"360%2A12+=+4320\"   DISABLED_style= \"max-width:90%; height:auto;\" > is my answer </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );