document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001499>Question 1001499</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A Discrete Mathematics student comes from the village of KILLYIKYAKYODLE. As an hilarious jape, on the way home from a late-night study session, she decides to rearrange the letters on one of the sign-posts.<BR>\n" );
document.write( "How many arrangements of the letters in KILLYIKYAKYODLE are possible?<BR>\n" );
document.write( "i got 15!/3!*2!*3!*2! for this<BR>\n" );
document.write( "Of these arrangements, how many have all the L’s together?<BR>\n" );
document.write( "Of these arrangements, how may have all the letters in alphabetical order? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618779 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618779\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> How many arrangements of the letters in KILLYIKYAKYODLE are possible?<BR>\n" );
document.write( "There are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K ALIGN=MIDDLE  ALT=\"K\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y ALIGN=MIDDLE  ALT=\"Y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I ALIGN=MIDDLE  ALT=\"I\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > each of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%7D%7D%7C+letters+%7B%7B%7BA ALIGN=MIDDLE  ALT=\"4%7D%7D%7C+letters+%7B%7B%7BA\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "for a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B3%2B3%2B2%2B4=15 ALIGN=MIDDLE  ALT=\"3%2B3%2B3%2B2%2B4=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > characters.<BR>\n" );
document.write( "If we could distinguish between the repeated instances of the same letter,<BR>\n" );
document.write( "we would see <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%21 ALIGN=MIDDLE  ALT=\"15%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > different arrangements.<BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K ALIGN=MIDDLE  ALT=\"K\"   DISABLED_style= \"max-width:90%; height:auto;\" >  can be arranged <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21=6 ALIGN=MIDDLE  ALT=\"3%21=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways.<BR>\n" );
document.write( "The same goes for<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y ALIGN=MIDDLE  ALT=\"Y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y ALIGN=MIDDLE  ALT=\"Y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "while the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I ALIGN=MIDDLE  ALT=\"I\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be arranged <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%21=2 ALIGN=MIDDLE  ALT=\"2%21=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways.<BR>\n" );
document.write( "So, if the repeats of the same letter cannot be distinguished, we would see<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815%21%2F%283%21%2A3%21%2A3%21%2A2%21%29%29 ALIGN=MIDDLE  ALT=\"highlight%2815%21%2F%283%21%2A3%21%2A3%21%2A2%21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > arrangements,<BR>\n" );
document.write( "which I believe is what you meant to write.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of these arrangements, how many have all the L’s together?<BR>\n" );
document.write( "If we do not include the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > instances of the letter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-3=12 ALIGN=MIDDLE  ALT=\"15-3=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > characters, that can be arranged into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%21%2F%283%21%2A3%21%2A2%21%29 ALIGN=MIDDLE  ALT=\"12%21%2F%283%21%2A3%21%2A2%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 12-letter \"words.\"<BR>\n" );
document.write( "In each 12-letter \"word\", we can then intercalate the 3 L’s together in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2B1=13 ALIGN=MIDDLE  ALT=\"12%2B1=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > different positions:<BR>\n" );
document.write( "before any of the 12 letter positions, or at the end.<BR>\n" );
document.write( "We end up with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2813%2A12%21%2F%283%21%2A3%21%2A2%21%29%29 ALIGN=MIDDLE  ALT=\"highlight%2813%2A12%21%2F%283%21%2A3%21%2A2%21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > arrangements that have all the L’s together.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of all the possible arrangements, how may have all the letters in alphabetical order?<BR>\n" );
document.write( "To have all the letters in alphabetical order,<BR>\n" );
document.write( "the first character must be A,<BR>\n" );
document.write( "the second character must be D,<BR>\n" );
document.write( "the third character must be E,<BR>\n" );
document.write( "the 2 I's must follow as the 4th and 5th characters,<BR>\n" );
document.write( "followed by the 3 K's,<BR>\n" );
document.write( "then the 3 L's,<BR>\n" );
document.write( "followed by the O,<BR>\n" );
document.write( "and ending with the 3 Y's.<BR>\n" );
document.write( "That is the prescribed order:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%29 ALIGN=MIDDLE  ALT=\"highlight%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > prescribed arrangement, no choices,<BR>\n" );
document.write( "unless we can distinguish between the repeated instances of the same letter. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );