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 #852299 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53763)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852299\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Three letters are randomly selected without replacement one at a time from {a,b,c,d----Z}.<BR>\n" );
document.write( "What is the probability that they are selected in alphabetical order. <BR>\n" );
document.write( "Express your answer as a common fraction. This was a question in 2010 state math counts<BR>\n" );
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document.write( "The set {a,b,c,d---Z) contains  26 + 26 = 52 letters.\r\n" );
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document.write( "Let's consider all possible triples of letters from this set without repeating.\r\n" );
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document.write( "Every triple has 6 incarnations (6 possible permutations).\r\n" );
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document.write( "Of these 6 permutations, ONLY ONE is in alphabetical order.\r\n" );
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document.write( "So, the probability under the problem's question is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   \r\n" );
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document.write( "It is the <U>ANSWER</U> to the problem.\r\n" );
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document.write( "Good problem and a nice solution.\r<BR>\n" );
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document.write( "I would say even more: this problem and the solution in some sense <BR>\n" );
document.write( "revolve your mind and teach you &nbsp;\" to think in the other way \".\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The reasoning can be constructed/re-told in other way, \r<BR>\n" );
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document.write( "&nbsp&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;giving \" another solution \".\r<BR>\n" );
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document.write( "The number of all ordered triples with no repeating is\r\n" );
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document.write( "    P(52,3) = 52*81*50.        (1)\r\n" );
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document.write( "The number of all unordered triples (=combinations) is\r\n" );
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document.write( "    C(52,3) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2852%2A51%2A50%29%2F%281%2A2%2A3%29 ALIGN=MIDDLE  ALT=\"%2852%2A51%2A50%29%2F%281%2A2%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.     (2)\r\n" );
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document.write( "Each single combination represents 6 permutations from P(52,3), which we count \r\n" );
document.write( "as one ordered triple in the alphabetical order.\r\n" );
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document.write( "Therefore, the seeking probability is the number of all different alphabetically ordered triples C(52,3)\r\n" );
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document.write( "to the number of all possible ordered triples P(52,3).\r\n" );
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document.write( "From formulas (1) and (2), it is clear that this ratio is\r\n" );
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document.write( "    P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2852%2C3%29%2FP%2852%2C3%29 ALIGN=MIDDLE  ALT=\"C%2852%2C3%29%2FP%2852%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "This completes the second solution.  We get the same answer.\r\n" );
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document.write( "The solution by @Theo is 73 lines long and they all go out the target.\r<BR>\n" );
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document.write( "His solution has many words, but is incorrect, and even has no right idea.\r<BR>\n" );
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document.write( "Now I want to present and to discuss another interesting fact.\r<BR>\n" );
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document.write( "Today, Oct.17, 2025, I submitted this problem to Google AI Overview, <BR>\n" );
document.write( "which works now as a universal free of charge solver to Math problems in the Internet.\r<BR>\n" );
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document.write( "Below is the Google AI Overview solution to this problem.<BR>\n" );
document.write( "I copy-pasted it today, Oct.17, 2025, 2:56 PM\r<BR>\n" );
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document.write( "- - - - - - - - AI Overview START - - - - - - - \r<BR>\n" );
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document.write( "The probability is 1/26*1/25*1/24, which is equal to 1/15600. \r\n" );
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document.write( "The total number of ways to select three letters in order is found \r\n" );
document.write( "by multiplying the number of options for each selection: 26*25*24 = 15600. \r\n" );
document.write( "Only one of these ways is in alphabetical order. \r\n" );
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document.write( "1. Total possible ordered selections: Since letters are selected without   \r\n" );
document.write( "   replacement, there are 26 choices for the first letter, 25 for the second,   \r\n" );
document.write( "   and 24 for the third. The total number of ordered combinations is . \r\n" );
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document.write( "2. Favorable outcome: There is only one way to select three letters that are in \r\n" );
document.write( "   alphabetical order (e.g., {a, b, c} in that specific order). \r\n" );
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document.write( "3. Calculate the probability: The probability is the number of favorable \r\n" );
document.write( "   outcomes divided by the total number of possible outcomes:  [1, 3, 4, 5, 6]  \r\n" );
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document.write( "AI responses may include mistakes.\r\n" );
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document.write( "[1] https://www.youtube.com/watch?v=7xu8kKVzT-w\r\n" );
document.write( "[2] https://brainly.com/question/30269108\r\n" );
document.write( "[3] https://gre.myprepclub.com/forum/three-letters-are-randomly-selected-one-at-a-time-without-replacement-23170.html\r\n" );
document.write( "[4] https://www.gauthmath.com/solution/1760404381135877/2-A-computer-randomly-selects-a-letter-from-the-alphabet-a-How-many-different-ou\r\n" );
document.write( "[5] https://study.com/skill/practice/calculating-probabilities-of-draws-without-replacement-questions.html\r\n" );
document.write( "[6] https://www.cliffsnotes.com/study-notes/24757581\r\n" );
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document.write( "            The link to this Google AI solution\r\n" );
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document.write( "https://www.google.com/search?q=Three+letters+are+randomly+selected+without+replacement+one+at+a+time+from+%7Ba%2Cb%2Cc%2Cd----Z%7D.+What+is+the+probability+that+they+are+selected+in+alphabetical+order.&rlz=1C1CHBF_enUS1071US1071&oq=Three+letters+are+randomly+selected+without+replacement+one+at+a+time+from+%7Ba%2Cb%2Cc%2Cd----Z%7D.+What+is+the+probability+that+they+are+selected+in+alphabetical+order.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIGCAEQRRg8MgYIAhBFGDzSAQkxODU4ajBqMTWoAgiwAgHxBZKTgCYYjWrL&sourceid=chrome&ie=UTF-8\r\n" );
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document.write( "- - - - - - - - AI Overview END - - - - - - - \r<BR>\n" );
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document.write( "Below is my (@ikleyn) comment.\r<BR>\n" );
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document.write( "This AI solution is FATALLY WRONG. It demonstrates complete and absolute misunderstanding of the subject.\r<BR>\n" );
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document.write( "So, it shows the level of authority of this solver.<BR>\n" );
document.write( "What is good that at the end of the solution, this AI solver honestly warns, that his output may include mistakes.<BR>\n" );
document.write( "This is exactly the case.\r<BR>\n" );
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document.write( "It also characterizes the level of the solutions in the Internet, on which Google AI Overview <BR>\n" );
document.write( "bases his conclusions. They also are at the zero level of understanding the subject.\r<BR>\n" );
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document.write( "Naturally, I reported to the Google AI Overview about this their mistake through their feedback system,<BR>\n" );
document.write( "and provided the link to this my post.\r<BR>\n" );
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document.write( "Hope, it will help to the AI to become a bit smarter.\r<BR>\n" );
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document.write( "In conclusion, couple more words to a reader.\r<BR>\n" );
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document.write( "A standard approach is this formal approach which is my solution #2.\r<BR>\n" );
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document.write( "Very often, it is the first move to many students to start thinking this way.\r<BR>\n" );
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document.write( "There is nothing bad in it, but it requires some level of maturity.<BR>\n" );
document.write( "Only those can go this way to the end, who are able \"to break through the wall <BR>\n" );
document.write( "with his/her forehead\", that is, with his/her mind.<BR>\n" );
document.write( "In other words, it requires certain level of training.\r<BR>\n" );
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document.write( "Therefore, if you have difficulties moving forward this way - - - try another way,<BR>\n" );
document.write( "as I did in my solution #1.<BR>\n" );
document.write( "May be, you intuition and/or your common sense will help you in this way.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is what I call &nbsp;\" to think differently \".\r<BR>\n" );
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document.write( " </SPAN>\n" );
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