Question 1208272
.
What is the number of different five-digit numbers that can be formed 
from the set S={2,3,4,5,6} such that one digit is repeated twice and another digit is repeated twice
~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
           <U>Solve step by step</U>



(a)  Let's consider more simple problem first:

         Given three different symbols A, B and C as an alphabet, 
         how many different 5-symbol words can be formed
         such that symbol A is used twice, symbol B is used twice 
         and symbol C is used once?


         For this problem, we can select a position for A in  {{{C[5]^2}}} = 10 ways;
                                         a position for B in  {{{C[3]^2}}} = 3 ways;
                                         then C occupies the remaining 5th position, at no choice.

         So, for this problems, there are 10*3 = 30 different words.



(b)  Now, returning to the given problem, we can assign any of 5 possible digit to A;
                                                        any of remaining 4 digits to B,
                                                    and any of remaining 3 digits to C.
  

     It gives 5*4*3 = 60 different choices for assigning; but for us, the pairs (A,B) and (B,A)
     are mutually-interchangeable : they lead to the same numbers.


     Therefore, this number 60 we should divide by 2 and take 60/2 = 30.



(c)  Now the final answer to the problem is  30 from (a) multiplied by 30 from (b)

     giving the  <U>ANSWER</U>  30 * 30 = 900.
</pre>

Solved.


------------------------


The solution and the final answer by &nbsp;Edwin are &nbsp;INCORRECT, &nbsp;since he considers / includes the pattern 
XXYYY with three repeating digits, &nbsp;which is &nbsp;PROHIBITED &nbsp;in this problem.


Saying that &nbsp;" When a digit is repeated three times, &nbsp;it is repeated two times ",
Edwin applies &nbsp;" logic imitation " to confuse you and to foist you that &nbsp;" 2 = 3 ", 
which is a dangerous &nbsp;(if not a fatal) &nbsp;delusion.


To ensure in it, &nbsp;give &nbsp;2 &nbsp;dollars to &nbsp;Edwin and request &nbsp;3 &nbsp;dollars back from him.


If he will protest, &nbsp;refer to his statement and say him that returning &nbsp;3 &nbsp;dollars, &nbsp;he returns &nbsp;2.



=======================



In the next his post, &nbsp;Edwin &nbsp;(as @mccravyedwin) &nbsp;makes self-justification for himself,
but it is, &nbsp;obviously, &nbsp;very crooked &nbsp;(toooo crooked) &nbsp;to be truth or to be accepted seriously.


It is only good to make you smile in response. 


Edwin, we all make mistakes from time to time.

Why do not you accept that you made a mistake ?



///////////////////////



An artificial Intelligence gives the same answer as mine (900 ways)


https://www.google.com/search?q=What+is+the+number+of+different+five-digit+numbers+that+can+be+formed+from+the+set+S%3D%7B2%2C3%2C4%2C5%2C6%7D+such+that+one+digit+is+repeated+twice+and+another+digit+is+repeated+twice&rlz=1C1CHBF_enUS1071US1071&oq=What+is+the+number+of+different+five-digit+numbers+that+can+be+formed+from+the+set+S%3D%7B2%2C3%2C4%2C5%2C6%7D+such+that+one+digit+is+repeated+twice+and+another+digit+is+repeated+twice&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTI5NzJqMGoxNagCCLACAQ&sourceid=chrome&ie=UTF-8


but with much less detailed explanations; so, you will not learn much from it.



\\\\\\\\\\\\\\\\\\\\\\\\\\



The other tutor expresses his doubts about clarity of wording.


To me, wording in this problem is PERFECT, as clear as the sun in sunny day, 
transparent as glass and does not allow any other/(any different) interpretation.



/\/\/\/\/\/\/\/\/\/\/\/



Regarding Edwin's (@AnlytcPhil) arguments about the meaning of words "repeating twice",
I do not think that the meaning is regulated by logic - it is regulated 
by commonly accepted rules of using words in specific area, by the context and by common sense.


For example, when a teacher says to a student "repeat it twice",
it clearly means that the teacher expects and the student should to pronounce
or to do something TWICE - not thrice.


Another example: when somebody says "I will not repeat it twice", it means
that the person, who said it, is going to pronounce his instruction only 
once - not three times.


So, I do not think that here the wording usage is determined by logic - it is determined 
by commonly accepted rules of using words in specific area, by the context and by the common sense.


Logic is IRRELEVANT here, especially logic, constructed artificially "ad hoc", not naturally.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Your argument, Edwin, in the last your post are pure speculation, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;logic is a kind of imitation and there is no the power of persuasion.



In general, further discussion of this issue becomes uninteresting, 
pointless and unproductive. It's like beating water in a mortar.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Dear colleagues Edwin and @greenestamps



Specially for our discussion, I performed a GOOGLE search.


My key words were "digit repeating twice",  and I selected only those GOOGLE outputs
that are/were relevant to our issue: what this wording means in Math ?


Please look into these 5 (five) sources/links below.


https://math.stackexchange.com/questions/3350036/probability-that-exactly-two-digits-repeat-twice-in-a-6-digit-number


https://www.youtube.com/watch?v=qfm6YPXBULE     


https://www.doubtnut.com/qna/648103004          

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;notice their language: how they say 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;"repeat digit twice" in their solution part 3 and how they interpret it.


https://infinitylearn.com/question-answer/the-total-number-of-three-digit-numbers-with-one-d-62cb5fabe06011cf7ead4d96


https://tardigrade.in/question/the-total-number-of-three-digit-numbers-with-one-digit-repeated-juypkept




In all these five source, the treatment of this passage "digit repeating twice" is the same as "digit appearing twice".



From it, I make a conclusion for myself that in Math 
"digit repeating twice" is the same as "digit appearing twice".



I know that around philologists in US 50 years ago was popular a prank "what does it means: two times appearing
or 3 times appearing", and they exercised and were drawn out quasi-logical proofs in both directions.


For philologists, it is normal. But mathematicians were NEVER engaged in such nonsense. 
For them, "repeating twice" always is the same as "appearing twice" - 
- it is a NORM of a mathematical language. 


This norm does exist in Math just hundreds of years, is commonly accepted everywhere in Math 
and by everybody in Math and is used without any discussions about its meaning.
It was obvious and clear to me from the very beginning, since I know the norms of the mathematical language.


Also, it is so in all quantitative sciences: Physics, Mechanics, Chemistry, Astronomy. 
In other areas, including philology or everyday life, it can be different, 
but again, the meaning by 60% or 80% or 90% or 99% depends on context - not on imitative logic.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So from this discussion, please draw a conclusion for yourself 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;that you did not know the correct usage and correct norm of usage

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of these words in Math until I explained it to you here today.



Why it was obvious and clear to me from the very beginning - because during my study at University,
during my work on my PhD and during the years of my professional work I read thousands articles 
and hundreds of books in English on Mathematics and on all kinds of Mechanics - so I know well this terminology.



From your side, it would be good and right to make two things:


&nbsp;&nbsp;&nbsp;&nbsp;(1)  &nbsp;&nbsp;To say "thanks" to me for my teaching, and


&nbsp;&nbsp;&nbsp;&nbsp;(2)  &nbsp;&nbsp;To remove all and any invectives to my side in your posts.


I sincerely expect that you will do it without delay and without reminders.



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



All nations on Earth have fairy tales for children, and many of these tales 
require the spell "door, open (!)" to be repeated three times.


How many times do you think the spell should be pronounced? Three times or four?
So, from childhood, children begin to understand what it means to repeat three times.


But you, under the influence of an old stupid philological prank, which happened 
50 years ago,  still believe that the spell should be said four times.
Well, the magic door will never open for you - until you change your beliefs from wrong to right.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Sept. 20, 2024.