document.write( "Question 481474: pls help me solve this problem in probability... thank you..\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "a ring of keys can hold 7 keys. In how many ways can the keys be arranged if the ring can be turned over? \n" ); document.write( "
Algebra.Com's Answer #852374 by ikleyn(52878)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( ".
\n" ); document.write( "pls help me solve this problem in probability... thank you..\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "a ring of keys can hold 7 keys. In how many ways can the keys be arranged if the ring can be turned over?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        The solution and the reasoning in the post by @Theo are incorrect.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        I came to bring a correct solution.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "The keys are on the ring - it means that we consider circular permutations of 7 keys.\r\n" );
document.write( "\r\n" );
document.write( "There are (7-1)! = 6! = 720 circular permutations of 7 keys.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, the ring can be turned over.  It means that we identify arrangements\r\n" );
document.write( "that differ by the mirror reflection.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we should divide 6! = 720 by 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The final ANSWER is: the number of all distinguishable arrangements is 720/2 = 360.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved correctly.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------------------\r
\n" ); document.write( "\n" ); document.write( "

The post-solution note

\r
\n" ); document.write( "\n" ); document.write( "        This problem is not from probability,  as you mistakenly write in your post.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        The area of  Math,  to which this problem does relate,  is called  Combinatorics.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );