document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164376>Question 1164376</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is more like math-language problem of mine rather than actual math.<BR>\n" );
document.write( "The question starts with \"In how many ways n women be seated in a row so that 2 PARTICULAR women will not be next to each other?\" Here, my understanding is that those 2 women can be treated as one entity and order does not matter ( Whether they sit Lt and Rt or Rt and Lt), and there is no need to invoke 2! ( I know that 2! is 2 and it does not matter here, but had it been 3 or more then whether it is expressed as 3! or 3 will matter, as you know).\r<BR>\n" );
document.write( "\n" );
document.write( "Then it goes on, as part of the solution, \"Wih no restrictions, n women may be seated in a row in nPn ways. I f 2 of the n women must always sit next to each other, the number of arrangements = 2!(n-1Pn-1).\" Here I think...although I am not sure...2! is appropriate, since there is no \"Particular\". If there was a \"Particular\"..2! will not be appropriate and only require 2. (Again please ref above regarding my comment about had it been >2)\r<BR>\n" );
document.write( "\n" );
document.write( "And then the solutions goes on to, \"Hence the number of ways n women can be seated in a row if 2 PARTICULAR women may never sit together = nPn - 2(n-1Pn-1).\" ...2 being the factorial of 2 in question.\r<BR>\n" );
document.write( "\n" );
document.write( "What happened it being \"Particular\" to start with, and how does substracting ( yes... once in a way I can't help using Chauser English for subtraction) arrangements of any two women equates (mathematically!) any two Particular women? \r<BR>\n" );
document.write( "\n" );
document.write( "Please dont tell me that I am hung up too much on those two PARTICULAR women !! (=<BR>\n" );
document.write( "Just tell me where I am getting turned around. I really need to understand this and get going so I dont insert or not insert n! inappropriately!\r<BR>\n" );
document.write( "\n" );
document.write( "Please help<BR>\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #788756 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52765)</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=788756\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>Solution</U>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "In all, there are n! different possible orderings/arrangements/permutations of \"n\" women in the row.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From this number, we should subtract the number of those permutations, where two women are seat together.\r\n" );
document.write( "\r\n" );
document.write( "The number of such permutations is 2*(n-1)!  (we threat this particular pair of women as one object and permutate then (n-1) objects; \r\n" );
document.write( "the multiplier 2 is to account for two different permutations of the type (Alicia,Barbara) and (Barbara,Alicia) inside this particular pair.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the final answer is  N = n! - 2*(n-1)!.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "-------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solved.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is a standard method solving this problem, and it is very close to your logic.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>Comment from student</U>: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "I understand that fully well. But my problem is the second paragraph: the number of arrangements = 2!(n-1Pn-1).\" \r\n" );
document.write( "Here I think...although I am not sure...2! is appropriate, since there is no \"Particular\". \r\n" );
document.write( "If there was a \"Particular\"..2! will not be appropriate and only require 2. \r\n" );
document.write( "\r\n" );
document.write( "(Again please ref above regarding my comment about had it been >2) So is the 2! is a typo in the book? \r\n" );
document.write( "\r\n" );
document.write( "I am a beginner. I am not willing to call anything in a math book a typo, until I check with a teacher; \r\n" );
document.write( "\r\n" );
document.write( "lest, >70% of answers will be typos for me!!(= Thank you\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>My response</U> :\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "You may treat the number of interior permutations of two particular women as 2! or 2.\r\n" );
document.write( "\r\n" );
document.write( "Since BOTH these values are the same, there is no \"error\" or \"typo\" in this case.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Did I fully answer your question ?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "/\/\/\/\/\/\/\\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>Comment from student</U> : \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Not really. It is ok that 2!=2 here and the answer did not change. My question is what if it was 3 PARTICULAR woman. \r\n" );
document.write( "Then why is it 3! since they are being treated as one entity? Obviously the answers are going to be different with 3 Vs 3!\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>My response</U> :  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "With 3 particular women the answer is  n! - 3!*(n-2)!   =    n! - 6*(n-2)!\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "And it is different problem then.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, I do not really understand your truble.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "/\/\/\/\/\/\/\/<BR>\n" );
document.write( "/\/\/\/\/\/\/\/\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>Comment from student</U>: When we are treating the 3 women as one entity when the order does not matter how they sit, <BR>\n" );
document.write( "why 3! (that leads to 6) ...why not just 3? 3! means that the 3 women can be seated in so (6) many different ways, <BR>\n" );
document.write( "which is not relevent if you are going to treat them sitting together as one entity...right ?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>My response</U> :  &nbsp;&nbsp;You ask me  &nbsp;&nbsp;\" . . . right ?\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "My response is &nbsp;&nbsp;\"WRONG\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We consider  these three woman as one object from the \"remoted point of view\";\r<BR>\n" );
document.write( "\n" );
document.write( "but we &nbsp;MUST &nbsp;ACCOUNT &nbsp;that &nbsp;&nbsp;3! = 6 interior permutations are possible inside this entity/object.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Honestly, &nbsp;&nbsp;I just tired to explain these simple things.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A prepared students must &nbsp;\"catch it\" &nbsp;from the air.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Have a nice day &nbsp;(!)  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\\\\\\\\\\\\\\\\\\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Yours misunderstanding starts from the point that you &nbsp;NEGLECT &nbsp;to read <BR>\n" );
document.write( "my wording solution from the beginning to the end.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you want to get understanding, &nbsp;you &nbsp;MUST &nbsp;read and &nbsp;RE-READ it until <BR>\n" );
document.write( "understanding will &nbsp;COME &nbsp;to your mind &nbsp;<U>suddenly</U> &nbsp;and &nbsp;<U>in full</U>.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I just did my part. &nbsp;Now &nbsp;YOU &nbsp;should do yours.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "/\/\/\/\/\/\/\/\/<BR>\n" );
document.write( "/\/\/\/\/\/\/\/\/<BR>\n" );
document.write( "/\/\/\/\/\/\/\/\/\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>Comment from student</U> : &nbsp;&nbsp;If the answer to my question was worded: \"The 2! is not there because they are sitting together, \r<BR>\n" );
document.write( "\n" );
document.write( "where 2 will suffice. It is there because the final answer demands not when they are sitting together, <BR>\n" );
document.write( "but when they are NOT sitting together.\" \r<BR>\n" );
document.write( "\n" );
document.write( "These two sentences would have put my question to rest, without needing any didactic explanations.! \r<BR>\n" );
document.write( "\n" );
document.write( "But never mind. I was thinking about it all night and I got it on my own. Now it will be etched for ever in my mind, <BR>\n" );
document.write( "essentially due to our tedious conversations! So all is not a waste, afterall...at leat for me!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<U>My response</U> :   &nbsp;&nbsp;My congrats &nbsp;(! ! !)    &nbsp;&nbsp;&nbsp;&nbsp;Finally &nbsp;(!)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Exactly as I said  &nbsp;&nbsp;\"If you want to get understanding, &nbsp;you &nbsp;MUST &nbsp;read and &nbsp;RE-READ it until <BR>\n" );
document.write( "understanding will &nbsp;COME &nbsp;to your mind &nbsp;<U>suddenly</U> &nbsp;and &nbsp;<U>in full</U>.\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I am very glad, &nbsp;at the end.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next time, &nbsp;when you want to have two-way conversation, &nbsp;please refer to the problem's &nbsp;ID &nbsp;number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I just explained to you once, &nbsp;what it is and where to find it, &nbsp;but I am not sure if you absorbed my explanations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );