document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=549349>Question 549349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three letters are randomly selected without replacement one at a time from {a,b,c,d----Z}.What is the probability that they are selected in alphabetical order. Express your answer as a common fraction. This was a question in 2010 state math counts  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #357671 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=357671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> there are 26 letters in the alphabet.<BR>\n" );
document.write( "probability of the first letter being in alphabetical order is 24 out of 26.<BR>\n" );
document.write( "last letter possible is letter x because y and z would come after that.<BR>\n" );
document.write( "y is no good because only 1 letter left after that in alphabetical.<BR>\n" );
document.write( "z is no good because no letters left after that in alphabetical order.<BR>\n" );
document.write( "probability of the second letter being in alphabetical order is 1 out of the remaining 25.<BR>\n" );
document.write( "probability of the last letter being in alphabeticak order is 1 out of the remaining 24.<BR>\n" );
document.write( "total probability is therefore 24/26 * 1/25 * 1/24<BR>\n" );
document.write( "to see how this works, assume only 5 letters.<BR>\n" );
document.write( "letters are a, b, c, d, e.<BR>\n" );
document.write( "probability here, if the formula is correct, would be 3/5 * 1/4 * 1/3 = 3/60<BR>\n" );
document.write( "let's see if that's correct.<BR>\n" );
document.write( "first letter could be a, b, or c<BR>\n" );
document.write( "if a, then b and c have to follow in that order.<BR>\n" );
document.write( "if b, then c and d have to follow in that order.<BR>\n" );
document.write( "if c, then d and e have to follow in that order.<BR>\n" );
document.write( "total possible combinations possible combinations of letters possible when order is important are:<BR>\n" );
document.write( "5P3 which comes out to be 5! / 2! which comes out to be (5*4*3*2*1) / (1*2) which comes out to be 60 possible permutations.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "abc acb bac bca cab cba<BR>\n" );
document.write( "abd adb bad bda dab dba<BR>\n" );
document.write( "abe aeb bae bea eab eba<BR>\n" );
document.write( "acd adc cad cda dac dca<BR>\n" );
document.write( "ace aec cae cea eac eca<BR>\n" );
document.write( "ade aed dae dea ead eda<BR>\n" );
document.write( "bcd bdc cbd cdb dbc dcb<BR>\n" );
document.write( "bce bec cbe ceb ebc ecb<BR>\n" );
document.write( "bde bed dbe deb ebd edb<BR>\n" );
document.write( "cde ced dce dec ecd edc<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "there should only be 3 out of the 60 that are in alphabetical order.<BR>\n" );
document.write( "they would be:<BR>\n" );
document.write( "abc<BR>\n" );
document.write( "bcd<BR>\n" );
document.write( "cde<BR>\n" );
document.write( "no others are possible.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since the same formula was applied to the 26 letters in the alphabet, then the answer for the alphabet should be as good.<BR>\n" );
document.write( "let's see if that's true.<BR>\n" );
document.write( "the probability for the alphabet was 24/26 * 1/25 * 1/24 which became (24*1*1) / (26*25*24) = 24 / 15600<BR>\n" );
document.write( "the number of possible permutations of 3 out of 26 is 26! / 23! = (26*25*24*23!) / 23! = (26*25*24) = 15600.<BR>\n" );
document.write( "out of these, the total number of possible sequences of letters in alphabetical order are:<BR>\n" );
document.write( "abc<BR>\n" );
document.write( "bcd<BR>\n" );
document.write( "cde<BR>\n" );
document.write( "def<BR>\n" );
document.write( "efg<BR>\n" );
document.write( "fgh<BR>\n" );
document.write( "ghi<BR>\n" );
document.write( "hij<BR>\n" );
document.write( "ijk<BR>\n" );
document.write( "jkl<BR>\n" );
document.write( "klm<BR>\n" );
document.write( "lmn<BR>\n" );
document.write( "mno<BR>\n" );
document.write( "nop<BR>\n" );
document.write( "opq<BR>\n" );
document.write( "pqr<BR>\n" );
document.write( "qrs<BR>\n" );
document.write( "rst<BR>\n" );
document.write( "stu<BR>\n" );
document.write( "tuv<BR>\n" );
document.write( "uvw<BR>\n" );
document.write( "vwx<BR>\n" );
document.write( "wxy<BR>\n" );
document.write( "xyz <BR>\n" );
document.write( "yz<BR>\n" );
document.write( "z<BR>\n" );
document.write( "you can see that only 24 out of the 26 letters of the alphabet have the possibility of being in alphabetical order 3 at a time.<BR>\n" );
document.write( "the total possible number of permutations of 3 out of 26 is given by the permutation formula of n! / (n-x)! where n is the total number of possible choices to pick from (26) and x is the number you pick at a time (3).<BR>\n" );
document.write( "26! / (26-3)! becomes 26! / 23! which is the number of 15600 we arrived at earlier.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );