Question 549349
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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
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<pre>
The set {a,b,c,d---Z) contains  26 + 26 = 52 letters.


Let's consider all possible triples of letters from this set without repeating.



Every triple has 6 incarnations (6 possible permutations).


Of these 6 permutations, ONLY ONE is in alphabetical order.


So, the probability under the problem's question is  {{{1/6}}}.   


It is the <U>ANSWER</U> to the problem.
</pre>

Good problem and a nice solution.


I would say even more: this problem and the solution in some sense 
revolve your mind and teach you &nbsp;" to think in the other way ".



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The reasoning can be constructed/re-told in other way, 

&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 ".



<pre>
The number of all ordered triples with no repeating is

    P(52,3) = 52*81*50.        (1)


The number of all unordered triples (=combinations) is

    C(52,3) = {{{(52*51*50)/(1*2*3)}}}.     (2)


Each single combination represents 6 permutations from P(52,3), which we count 
as one ordered triple in the alphabetical order.


Therefore, the seeking probability is the number of all different alphabetically ordered triples C(52,3)

to the number of all possible ordered triples P(52,3).


From formulas (1) and (2), it is clear that this ratio is

    P = {{{C(52,3)/P(52,3)}}} = {{{1/6}}}.


This completes the second solution.  We get the same answer.
</pre>


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The solution by @Theo is 73 lines long and they all go out the target.


His solution has many words, but is incorrect, and even has no right idea.



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Now I want to present and to discuss another interesting fact.


Today, Oct.17, 2025, I submitted this problem to Google AI Overview, 
which works now as a universal free of charge solver to Math problems in the Internet.


Below is the Google AI Overview solution to this problem.
I copy-pasted it today, Oct.17, 2025, 2:56 PM


- - - - - - - - AI Overview START - - - - - - - 


<pre>
The probability is 1/26*1/25*1/24, which is equal to 1/15600. 

The total number of ways to select three letters in order is found 
by multiplying the number of options for each selection: 26*25*24 = 15600. 
Only one of these ways is in alphabetical order. 

1. Total possible ordered selections: Since letters are selected without   
   replacement, there are 26 choices for the first letter, 25 for the second,   
   and 24 for the third. The total number of ordered combinations is . 

2. Favorable outcome: There is only one way to select three letters that are in 
   alphabetical order (e.g., {a, b, c} in that specific order). 

3. Calculate the probability: The probability is the number of favorable 
   outcomes divided by the total number of possible outcomes:  [1, 3, 4, 5, 6]  

AI responses may include mistakes.

[1] https://www.youtube.com/watch?v=7xu8kKVzT-w
[2] https://brainly.com/question/30269108
[3] https://gre.myprepclub.com/forum/three-letters-are-randomly-selected-one-at-a-time-without-replacement-23170.html
[4] https://www.gauthmath.com/solution/1760404381135877/2-A-computer-randomly-selects-a-letter-from-the-alphabet-a-How-many-different-ou
[5] https://study.com/skill/practice/calculating-probabilities-of-draws-without-replacement-questions.html
[6] https://www.cliffsnotes.com/study-notes/24757581


            The link to this Google AI solution

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
</pre>

- - - - - - - - AI Overview END - - - - - - - 



Below is my (@ikleyn) comment.


This AI solution is FATALLY WRONG. It demonstrates complete and absolute misunderstanding of the subject.


So, it shows the level of authority of this solver.
What is good that at the end of the solution, this AI solver honestly warns, that his output may include mistakes.
This is exactly the case.


It also characterizes the level of the solutions in the Internet, on which Google AI Overview 
bases his conclusions. They also are at the zero level of understanding the subject.



Naturally, I reported to the Google AI Overview about this their mistake through their feedback system,
and provided the link to this my post.


Hope, it will help to the AI to become a bit smarter.



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In conclusion, couple more words to a reader.


A standard approach is this formal approach which is my solution #2.


Very often, it is the first move to many students to start thinking this way.


There is nothing bad in it, but it requires some level of maturity.
Only those can go this way to the end, who are able "to break through the wall 
with his/her forehead", that is, with his/her mind.
In other words, it requires certain level of training.


Therefore, if you have difficulties moving forward this way - - - try another way,
as I did in my solution #1.
May be, you intuition and/or your common sense will help you in this way.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is what I call &nbsp;" to think differently ".