document.write( "Question 1119602: There are 6 people who will sit in a row but out of them James will always be left of Esther and John will always be right of Esther. In how many ways such arrangement can be done?\r
\n" ); document.write( "\n" ); document.write( "Note: i tried solving and got 24 as the answer but there was no 24 in the options. here are the options: \r
\n" ); document.write( "\n" ); document.write( "120
\n" ); document.write( "72
\n" ); document.write( "64
\n" ); document.write( "218.\r
\n" ); document.write( "\n" ); document.write( "Many thanks. \n" ); document.write( "

Algebra.Com's Answer #735195 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "I will provide here another solution.
\n" ); document.write( "Probably, it is that \"elegant\" solution John mentioned in his post.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "For each given arrangement from the condition, consider all 6 (six) permutations of the three people James, Esther and John \r\n" );
document.write( "inside of the greater arrangement.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this way, you will get all possible 6! = 1*2*3*4*5*6 = 720 arrangements of 6 people in a row.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the number of all arrangements under the problem's question is   = 120.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Answer.  The number of all arrangements under the problem's question is  120.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-------------------
\n" ); document.write( "Comment from student: thank you so much Ikleyn. i so much appreciate. i have liked and recommended your page to my friends. \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "pls i still have one more. \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Dada want to change her password which is dada112233 but with same letters and number. In how many ways she can do that? thank you in anticipation.
\n" ); document.write( "-------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "My responce. Although it is not formulated explicitly and directly in the condition, I will assume that\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    4 letters occupy 4 first positions,  and\r\n" );
document.write( "\r\n" );
document.write( "    6 digits occupy the last 6 positions.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "So, we have all distinguishable arrangements of the word \"dada\" in the first 4 positions and all distinguishable arrangements \r\n" );
document.write( "\r\n" );
document.write( "of 6 digits \"112233\" in the last 6 positions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The word \"dada\" has 4 letters in all; of them, there are only 2 distinguishable letters each of the multiplicity 2.\r\n" );
document.write( "\r\n" );
document.write( "The number of all distinguishable arrangements for letters is   =  = 6 in this case.\r\n" );
document.write( "\r\n" );
document.write( "    (2!*2!) in the denominator stays to account for repeated \"a\" and repeated \"d\" with their multiplicities.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The word \"112233\" has 6 digits in all; of them, there are only 3 distinguishable digits  each of the multiplicity 2.\r\n" );
document.write( "\r\n" );
document.write( "The number of all distinguishable arrangements for digits is   =  = 90 in this case.\r\n" );
document.write( "\r\n" );
document.write( "    (2!*2!*2!) in the denominator stays to account for repeated \"1\"; repeated \"2\", and repeated \"3\" with their multiplicities.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, to complete the solution, we need simply multiply 6 by 90.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Answer.  There are 540 different ways to create the password in a way described in the problem.

\r
\n" ); document.write( "\n" ); document.write( "See the lesson\r
\n" ); document.write( "\n" ); document.write( " - Arranging elements of sets containing indistinguishable elements \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic \"Combinatorics: Combinations and permutations\". \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "=============\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    In the future, when you post your message through the \"Thank you/comment\" window, following my post, please refer\r\n" );
document.write( "\r\n" );
document.write( "    to the problem ID number (which is  \"1119602\"  in this case)\r\n" );
document.write( "\r\n" );
document.write( "    in order I could identify the problem properly and answer under the appropriate post.  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thank you.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );