document.write( "Question 1082578: How many distinct ways can the letters in the word ITEMS be arranged?
\n" ); document.write( "How many distinct ways can the letters in the word STEMS be arranged?
\n" ); document.write( "How many distinct ways can the letters in the word SEEMS be arranged?
\n" ); document.write( "What makes these counts different? \n" ); document.write( "

Algebra.Com's Answer #696739 by ikleyn(52803) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "1) How many distinct ways can the letters in the word ITEMS be arranged?
\n" ); document.write( "2) How many distinct ways can the letters in the word STEMS be arranged?
\n" ); document.write( "3) How many distinct ways can the letters in the word SEEMS be arranged?
\n" ); document.write( "What makes these counts different?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "1) The word ITEMS has 5 letters. They all are different (distinguishable).\r\n" );
document.write( "\r\n" );
document.write( "   Therefore, there are 5! = 5*4*3*2*1 = 120 distinct ways the letters in the word ITEMS can be arranged.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2)  The word STEMS has 5 letters. \r\n" );
document.write( "\r\n" );
document.write( "    There are 4 and only 4 different (distinguishable) letters. Two letters (S) are identical.\r\n" );
document.write( "\r\n" );
document.write( "     Although there are formally 5! = 120 permutations/arrangements, not all of them are distinct/distinguishable.\r\n" );
document.write( "\r\n" );
document.write( "     Namely, in each permutation two identical letters S can be reversed in their positions, but the resulting permutations still represent the same arrangement.\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the whole number of permutations must be divided by 2 to account for this fact.\r\n" );
document.write( "\r\n" );
document.write( "     As a result, the final formula for the number of arrangements in this case is   = 60.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3)  The word SEEMS has 5 letters. \r\n" );
document.write( "\r\n" );
document.write( "    There are 3 and only 3 different (distinguishable) letters. There are two identical letters S and two identical letters E.\r\n" );
document.write( "\r\n" );
document.write( "     Following to the logic of the n 2), we must divide 120 (the total number of formal permutations of 5 symbols) by (2*2) = 4.\r\n" );
document.write( "\r\n" );
document.write( "     As a result, the final formula for the number of arrangements in this case is   = 30.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "All question are answered. The problem is solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On permutations, see the lessons \r
\n" ); document.write( "\n" ); document.write( "    - Introduction to Permutations\r
\n" ); document.write( "\n" ); document.write( "    - PROOF of the formula on the number of Permutations\r
\n" ); document.write( "\n" ); document.write( "    - Problems on Permutations\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\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 lessons are 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( "

\n" );