document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208272>Question 1208272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846583 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</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=846583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "What is the number of different five-digit numbers that can be formed <BR>\n" );
document.write( "from the set S={2,3,4,5,6} such that one digit is repeated twice and another digit is repeated twice<BR>\n" );
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document.write( "           <U>Solve step by step</U>\r\n" );
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document.write( "(a)  Let's consider more simple problem first:\r\n" );
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document.write( "         Given three different symbols A, B and C as an alphabet, \r\n" );
document.write( "         how many different 5-symbol words can be formed\r\n" );
document.write( "         such that symbol A is used twice, symbol B is used twice \r\n" );
document.write( "         and symbol C is used once?\r\n" );
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document.write( "         For this problem, we can select a position for A in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5B5%5D%5E2 ALIGN=MIDDLE  ALT=\"C%5B5%5D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 10 ways;\r\n" );
document.write( "                                         a position for B in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5B3%5D%5E2 ALIGN=MIDDLE  ALT=\"C%5B3%5D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3 ways;\r\n" );
document.write( "                                         then C occupies the remaining 5th position, at no choice.\r\n" );
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document.write( "         So, for this problems, there are 10*3 = 30 different words.\r\n" );
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document.write( "(b)  Now, returning to the given problem, we can assign any of 5 possible digit to A;\r\n" );
document.write( "                                                        any of remaining 4 digits to B,\r\n" );
document.write( "                                                    and any of remaining 3 digits to C.\r\n" );
document.write( "  \r\n" );
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document.write( "     It gives 5*4*3 = 60 different choices for assigning; but for us, the pairs (A,B) and (B,A)\r\n" );
document.write( "     are mutually-interchangeable : they lead to the same numbers.\r\n" );
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document.write( "     Therefore, this number 60 we should divide by 2 and take 60/2 = 30.\r\n" );
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document.write( "(c)  Now the final answer to the problem is  30 from (a) multiplied by 30 from (b)\r\n" );
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document.write( "     giving the  <U>ANSWER</U>  30 * 30 = 900.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "The solution and the final answer by &nbsp;Edwin are &nbsp;INCORRECT, &nbsp;since he considers / includes the pattern <BR>\n" );
document.write( "XXYYY with three repeating digits, &nbsp;which is &nbsp;PROHIBITED &nbsp;in this problem.\r<BR>\n" );
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document.write( "Saying that &nbsp;\" When a digit is repeated three times, &nbsp;it is repeated two times \",<BR>\n" );
document.write( "Edwin applies &nbsp;\" logic imitation \" to confuse you and to foist you that &nbsp;\" 2 = 3 \", <BR>\n" );
document.write( "which is a dangerous &nbsp;(if not a fatal) &nbsp;delusion.\r<BR>\n" );
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document.write( "To ensure in it, &nbsp;give &nbsp;2 &nbsp;dollars to &nbsp;Edwin and request &nbsp;3 &nbsp;dollars back from him.\r<BR>\n" );
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document.write( "If he will protest, &nbsp;refer to his statement and say him that returning &nbsp;3 &nbsp;dollars, &nbsp;he returns &nbsp;2.\r<BR>\n" );
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document.write( "In the next his post, &nbsp;Edwin &nbsp;(as @mccravyedwin) &nbsp;makes self-justification for himself,<BR>\n" );
document.write( "but it is, &nbsp;obviously, &nbsp;very crooked &nbsp;(toooo crooked) &nbsp;to be truth or to be accepted seriously.\r<BR>\n" );
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document.write( "It is only good to make you smile in response. \r<BR>\n" );
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document.write( "Edwin, we all make mistakes from time to time.\r<BR>\n" );
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document.write( "Why do not you accept that you made a mistake ?\r<BR>\n" );
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document.write( "An artificial Intelligence gives the same answer as mine (900 ways)\r<BR>\n" );
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document.write( "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\r<BR>\n" );
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document.write( "but with much less detailed explanations; so, you will not learn much from it.\r<BR>\n" );
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document.write( "The other tutor expresses his doubts about clarity of wording.\r<BR>\n" );
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document.write( "To me, wording in this problem is PERFECT, as clear as the sun in sunny day, <BR>\n" );
document.write( "transparent as glass and does not allow any other/(any different) interpretation.\r<BR>\n" );
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document.write( "Regarding Edwin's (@AnlytcPhil) arguments about the meaning of words \"repeating twice\",<BR>\n" );
document.write( "I do not think that the meaning is regulated by logic - it is regulated <BR>\n" );
document.write( "by commonly accepted rules of using words in specific area, by the context and by common sense.\r<BR>\n" );
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document.write( "For example, when a teacher says to a student \"repeat it twice\",<BR>\n" );
document.write( "it clearly means that the teacher expects and the student should to pronounce<BR>\n" );
document.write( "or to do something TWICE - not thrice.\r<BR>\n" );
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document.write( "Another example: when somebody says \"I will not repeat it twice\", it means<BR>\n" );
document.write( "that the person, who said it, is going to pronounce his instruction only <BR>\n" );
document.write( "once - not three times.\r<BR>\n" );
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document.write( "So, I do not think that here the wording usage is determined by logic - it is determined <BR>\n" );
document.write( "by commonly accepted rules of using words in specific area, by the context and by the common sense.\r<BR>\n" );
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document.write( "Logic is IRRELEVANT here, especially logic, constructed artificially \"ad hoc\", not naturally.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Your argument, Edwin, in the last your post are pure speculation, <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;logic is a kind of imitation and there is no the power of persuasion.\r<BR>\n" );
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document.write( "In general, further discussion of this issue becomes uninteresting, <BR>\n" );
document.write( "pointless and unproductive. It's like beating water in a mortar.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Dear colleagues Edwin and @greenestamps\r<BR>\n" );
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document.write( "Specially for our discussion, I performed a GOOGLE search.\r<BR>\n" );
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document.write( "My key words were \"digit repeating twice\",  and I selected only those GOOGLE outputs<BR>\n" );
document.write( "that are/were relevant to our issue: what this wording means in Math ?\r<BR>\n" );
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document.write( "Please look into these 5 (five) sources/links below.\r<BR>\n" );
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document.write( "https://math.stackexchange.com/questions/3350036/probability-that-exactly-two-digits-repeat-twice-in-a-6-digit-number\r<BR>\n" );
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document.write( "https://www.youtube.com/watch?v=qfm6YPXBULE     \r<BR>\n" );
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document.write( "https://www.doubtnut.com/qna/648103004          \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;notice their language: how they say <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\"repeat digit twice\" in their solution part 3 and how they interpret it.\r<BR>\n" );
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document.write( "https://infinitylearn.com/question-answer/the-total-number-of-three-digit-numbers-with-one-d-62cb5fabe06011cf7ead4d96\r<BR>\n" );
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document.write( "https://tardigrade.in/question/the-total-number-of-three-digit-numbers-with-one-digit-repeated-juypkept\r<BR>\n" );
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document.write( "In all these five source, the treatment of this passage \"digit repeating twice\" is the same as \"digit appearing twice\".\r<BR>\n" );
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document.write( "From it, I make a conclusion for myself that in Math <BR>\n" );
document.write( "\"digit repeating twice\" is the same as \"digit appearing twice\".\r<BR>\n" );
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document.write( "I know that around philologists in US 50 years ago was popular a prank \"what does it means: two times appearing<BR>\n" );
document.write( "or 3 times appearing\", and they exercised and were drawn out quasi-logical proofs in both directions.\r<BR>\n" );
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document.write( "For philologists, it is normal. But mathematicians were NEVER engaged in such nonsense. <BR>\n" );
document.write( "For them, \"repeating twice\" always is the same as \"appearing twice\" - <BR>\n" );
document.write( "- it is a NORM of a mathematical language. \r<BR>\n" );
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document.write( "This norm does exist in Math just hundreds of years, is commonly accepted everywhere in Math <BR>\n" );
document.write( "and by everybody in Math and is used without any discussions about its meaning.<BR>\n" );
document.write( "It was obvious and clear to me from the very beginning, since I know the norms of the mathematical language.\r<BR>\n" );
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document.write( "Also, it is so in all quantitative sciences: Physics, Mechanics, Chemistry, Astronomy. <BR>\n" );
document.write( "In other areas, including philology or everyday life, it can be different, <BR>\n" );
document.write( "but again, the meaning by 60% or 80% or 90% or 99% depends on context - not on imitative logic.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So from this discussion, please draw a conclusion for yourself \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;that you did not know the correct usage and correct norm of usage\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of these words in Math until I explained it to you here today.\r<BR>\n" );
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document.write( "Why it was obvious and clear to me from the very beginning - because during my study at University,<BR>\n" );
document.write( "during my work on my PhD and during the years of my professional work I read thousands articles <BR>\n" );
document.write( "and hundreds of books in English on Mathematics and on all kinds of Mechanics - so I know well this terminology.\r<BR>\n" );
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document.write( "From your side, it would be good and right to make two things:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;(1)  &nbsp;&nbsp;To say \"thanks\" to me for my teaching, and\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;(2)  &nbsp;&nbsp;To remove all and any invectives to my side in your posts.\r<BR>\n" );
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document.write( "I sincerely expect that you will do it without delay and without reminders.\r<BR>\n" );
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document.write( "All nations on Earth have fairy tales for children, and many of these tales <BR>\n" );
document.write( "require the spell \"door, open (!)\" to be repeated three times.\r<BR>\n" );
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document.write( "How many times do you think the spell should be pronounced? Three times or four?<BR>\n" );
document.write( "So, from childhood, children begin to understand what it means to repeat three times.\r<BR>\n" );
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document.write( "But you, under the influence of an old stupid philological prank, which happened <BR>\n" );
document.write( "50 years ago,  still believe that the spell should be said four times.<BR>\n" );
document.write( "Well, the magic door will never open for you - until you change your beliefs from wrong to right.\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;&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.\r<BR>\n" );
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