document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202976>Question 1202976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An absent-minded secretary randomly puts four letters into four envelopes. What is the probability that at least one of the letters will be in the correct envelopes. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838197 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838197\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The response from the other tutor arrives at the correct answer only by chance; the method used is faulty.  If we use that method on the problem with only 3 envelopes, the answer we get is<br><BR>\n" );
document.write( "3C1/3!+3C2/3!+3C3/3! = 3/6 + 3/6 + 1/6 = 7/6<br><BR>\n" );
document.write( "which of course is not a valid number for a probability.<br><BR>\n" );
document.write( "The probability that at least one letter gets into the right envelope is 1, minus the probability that none of them get in the right envelope.<br><BR>\n" );
document.write( "If no letter gets in the right envelope, we have a derangement -- an arrangement in which none of the letters are in the right place.  The number of derangements of 4 items is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21%281-1%2F1%21%2B1%2F2%21-1%2F3%21%2B1%2F4%21%29 ALIGN=MIDDLE  ALT=\"4%21%281-1%2F1%21%2B1%2F2%21-1%2F3%21%2B1%2F4%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%281-1%2B1%2F2-1%2F6%2B1%2F24%29 ALIGN=MIDDLE  ALT=\"24%281-1%2B1%2F2-1%2F6%2B1%2F24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-4%2B1+=+9 ALIGN=MIDDLE  ALT=\"12-4%2B1+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(N.B. -- Do an internet search on \"derangement\" if you want to try to understand that formula....)<br><BR>\n" );
document.write( "Since there are 4! = 24 possible arrangements of the 4 letters, the probability that we have a derangement (none of the letters is in the right envelope) is 9/24 = 3/8<br><BR>\n" );
document.write( "So the probability that at least one letter gets in the right envelope is 1-3/8 = 5/8.<br><BR>\n" );
document.write( "ANSWER: 5/8<br><BR>\n" );
document.write( " </SPAN>\n" );
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