document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=880687>Question 880687</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A three-letter code is generated using the letters of the alphabet. How many of these codes contain at least 1 vowel? The answer is apparently 7620. Does order matter? Are repetitions allowed?\r<BR>\n" );
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document.write( "This is what I have tried so far:\r<BR>\n" );
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document.write( "At least 1 vowel means that the 3 letter code can either contain 1 vowel, 2 vowels, or 3 vowels so I set up my equation like this:<BR>\n" );
document.write( "(5C1 * 21C2) + (5C2 * 21C1) + (5C3 * 21C0) = 1270 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #531645 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=531645\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You used combinations rather than permutations;<BR>\n" );
document.write( "so you are finding the number of sets (order does not matter) of 3 letters including at least one vowel, no repetitions allowed.<BR>\n" );
document.write( "One of your sets is {A,B,C}.<BR>\n" );
document.write( "However, the set {A,B,C} is shared by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > different codes.<BR>\n" );
document.write( "Codes ABC, ACB, BAC, BCA, CAB, and CBA are considered different codes, because they are different permutations of the elements of the set, and for a code order matters.<BR>\n" );
document.write( "Each of your <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1270 ALIGN=MIDDLE  ALT=\"1270\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 3 letters generates <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;\" > codes,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1270%2A6=7620 ALIGN=MIDDLE  ALT=\"1270%2A6=7620\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The result <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7620 ALIGN=MIDDLE  ALT=\"7620\"   DISABLED_style= \"max-width:90%; height:auto;\" > requires that it be specified that repetitions are not allowed. (If repetitions were allowed, as usually happens for codes< the result would be a higher number, but the problem would not be so interesting).<BR>\n" );
document.write( "That result <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7620 ALIGN=MIDDLE  ALT=\"7620\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be reached most directly by<BR>\n" );
document.write( "subtracting the number of 3-consonant no-repeat codes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2A20%2A19=7980 ALIGN=MIDDLE  ALT=\"21%2A20%2A19=7980\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "from the number of 3-letter no-repeat codes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26%2A25%2A24=15600 ALIGN=MIDDLE  ALT=\"26%2A25%2A24=15600\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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