document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=588172>Question 588172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A security code consists of four different letters chosen from the 26 in alphabet, order being important. For example,  QCDK. Given that the code does not contain any vowels what is the probability that it will contain an X?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #374420 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=374420\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ONE WAY:<BR>\n" );
document.write( "I would calculate the probability of no X and subtract from 1.<BR>\n" );
document.write( "Not counting Y as a vowel, there are 5 vowels in the 26 alphabet letters, leaving 21 consonants to choose from.<BR>\n" );
document.write( "The probability of not having an X as the first letter would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2F21 ALIGN=MIDDLE  ALT=\"20%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The probability of not having an X as the second letter would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F20 ALIGN=MIDDLE  ALT=\"19%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because whatever letter was chosen first would not be available.<BR>\n" );
document.write( "The probability of not having an X as the third letter would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F19 ALIGN=MIDDLE  ALT=\"18%2F19\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The probability of not having an X as the fourth letter would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%2F18 ALIGN=MIDDLE  ALT=\"17%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The probability of not having an X would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820%2F21%29%2819%2F20%29%2818%2F19%29%2817%2F18%29=20%2A19%2A18%2A17%2F%2821%2A20%2A19%2A18%29=17%2F21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The probability of having one X would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-17%2F21=4%2F21 ALIGN=MIDDLE  ALT=\"1-17%2F21=4%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "ANOTHER WAY:<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2A19%2A18 ALIGN=MIDDLE  ALT=\"20%2A19%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to choose a 3-letter combination with no repeated letters from the set of consonants that are not X.<BR>\n" );
document.write( "Inserting an X in any of the 4 possible positions, you get the<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A20%2A19%2A18 ALIGN=MIDDLE  ALT=\"4%2A20%2A19%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" > combinations of four different consonants including one X.<BR>\n" );
document.write( "There is a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2A20%2A19%2A18 ALIGN=MIDDLE  ALT=\"21%2A20%2A19%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" > combinations of four different consonants.<BR>\n" );
document.write( "Among those, we had found that only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A20%2A19%2A18 ALIGN=MIDDLE  ALT=\"4%2A20%2A19%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" > could include an X.<BR>\n" );
document.write( "The fraction including an X is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A20%2A19%2A18%2F%2821%2A20%2A19%2A18%29=4%2F21 ALIGN=MIDDLE  ALT=\"4%2A20%2A19%2A18%2F%2821%2A20%2A19%2A18%29=4%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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