document.write( "Question 1210183: In how many ways can we seat 3 pairs of siblings in a row of 10 chairs, so that nobody sits next to their sibling? (Two chairs will be left empty, of course.)
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #851658 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "In how many ways can we seat 3 pairs of siblings in a row of 10 chairs, so that nobody sits next to their sibling?
\n" ); document.write( "(Two chairs will be left empty, of course.)
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Before to start, I want to make couple of notices.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        First, this instruction \" Two chairs will be left empty, of course \" is, OBVIOUSLY, incorrect.
\n" ); document.write( "        The correct instruction should say \" Four chairs will be left empty, of course \".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Second, the meaning of the problem is that for each pair of siblings, the paired siblings do not seat in adjacent chairs. \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This problem can be solved using Inclusion-Exclusion principle step by step, following this logic.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let's the pairs of siblings be A=(x,y), B=(u,v), and C=(z,t)  (so, A, B and C are their family names).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 1.  First, we consider all possible different placements of 6 persons (three pairs of siblings) in 10 chairs \r\n" );
document.write( "         without any constraints.\r\n" );
document.write( "\r\n" );
document.write( "         The number of these placements is 10*9*8*7*6*5 = 151200, so we consider the list of all such 151200 placements.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 2.  From this list, we cross out all arrangements, where (x,y) are next to each other.\r\n" );
document.write( "\r\n" );
document.write( "         We consider the pair (x,y) as one unit; so, we have this unit and 4 other persons to arrange them \r\n" );
document.write( "         in 9 places. It can be done by N1 = 9*(8*7*6*5) different ways. It means that for (x,y) we cross out  \r\n" );
document.write( "         N1 =(9)*(8*7*6*5) = 15120 records from the list of all 151200 placements.\r\n" );
document.write( "\r\n" );
document.write( "         We do the same for the pairs (y,x), (u,v), (v,u), (z,t) and (t,z).\r\n" );
document.write( "\r\n" );
document.write( "         Thus, doing this way, we cross out, in all, 6*N1 similar records from the list .\r\n" );
document.write( "\r\n" );
document.write( "         After this step, we have 151200-6*N1 = 151200 - 6*15120 = 60480 records in the list.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "                  Again, for now, there are  60480  records in our list.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 3.  But doing it, we cross out some records several times.  \r\n" );
document.write( "         We cross out several time the records, where we find \r\n" );
document.write( "\r\n" );
document.write( "    - the pair (x,y) sitting together and the pair (u,v) sitting together;\r\n" );
document.write( "    - the pair (x,y) sitting together and the pair (z,t) sitting together;\r\n" );
document.write( "    - the pair (u,v) sitting together and the pair (z,t) sitting together.\r\n" );
document.write( "\r\n" );
document.write( "    - the pair (y,x) sitting together and the pair (u,v) sitting together;\r\n" );
document.write( "    - the pair (y,x) sitting together and the pair (z,t) sitting together;\r\n" );
document.write( "    - the pair (v,u) sitting together and the pair (z,t) sitting together.\r\n" );
document.write( "\r\n" );
document.write( "    - the pair (x,y) sitting together and the pair (v,u) sitting together;\r\n" );
document.write( "    - the pair (x,y) sitting together and the pair (t,z) sitting together;\r\n" );
document.write( "    - the pair (u,v) sitting together and the pair (z,t) sitting together.\r\n" );
document.write( "\r\n" );
document.write( "    - the pair (y,x) sitting together and the pair (v,u) sitting together;\r\n" );
document.write( "    - the pair (y,x) sitting together and the pair (t,z) sitting together;\r\n" );
document.write( "    - the pair (v,u) sitting together and the pair (t,z) sitting together.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "              So, we want to know the number of such records what we crossed out \r\n" );
document.write( "                     several times to compensate them in the future.\r\n" );
document.write( "                                Therefore, we make \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 4.  Let's determine the number  N2 of all possible placements of pair A and B, \r\n" );
document.write( "         where siblings (x,y) seat next to each other, and siblings (u,v) seat next to each other,\r\n" );
document.write( "         without other restrictions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         For it, we consider the pair (x,y) as one unit and consider the pair (u,v) as the other unit. \r\n" );
document.write( "         so, we have these two units and 2 other persons to arrange them in (10-1-1) = 8 places. \r\n" );
document.write( "         It can be done by N2= (8*7)*(6*5) = 1680 different ways. \r\n" );
document.write( "\r\n" );
document.write( "         It means that in the step 3 we crossed out N2 = (8*7)*(6*5) = 1680 records 12 times from the list\r\n" );
document.write( "         of all 151200 placements.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         So, to compensate multiply crossing, we should restore  12*N2 = 12*1680 = 20160  records \r\n" );
document.write( "         in the list of  placements.\r\n" );
document.write( "\r\n" );
document.write( "         \r\n" );
document.write( "                 Thus, for now, the list of records will contain  60480 + 20160 = 80640  records. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 5.  But again, doing this way, we restore some records in the list several times.\r\n" );
document.write( "\r\n" );
document.write( "         Namely, we restore several times the records, where siblings (x,y) seat together,\r\n" );
document.write( "         as well as siblings (u,v) seat together and siblings (z,t) seat together.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         Let's calculate, in how many cases N2 siblings (x,y) seat together, as well as siblings (u,v)seat together\r\n" );
document.write( "         and siblings (z,t) seat together in 10 chairs in the row.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         So, we consider three pairs A = (x,y), B = (u,v) and C = (z,t) as three units, and we can place them in (10-3) = 7 positions.\r\n" );
document.write( "\r\n" );
document.write( "         It can be done in  N2 = 7*6*5 = 210 different ways.\r\n" );
document.write( "\r\n" );
document.write( "         In these records, there are 2 possible permutations of siblings inside each pair.\r\n" );
document.write( "\r\n" );
document.write( "         So, the total number of these repeating restoring is   = 8*N2 = 8*(7*6*5) = 8*210 = 1680.\r\n" );
document.write( "\r\n" );
document.write( "         Hence, to keep the balance, we should subtract 1680 from 80640.\r\n" );
document.write( "         \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Step 6.  Thus, the final number of arranging/placing of three pairs of siblings in accordance with the problem is\r\n" );
document.write( "\r\n" );
document.write( "             80640 - 1680 = 78960.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the problem is solved completely.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  There are 78960 different ways to arrange 3 pairs of siblings in a row of 10 chairs\r\n" );
document.write( "         such that, in paired siblings, nobody sits next to their sibling.\r\n" );
document.write( "

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

\n" );