document.write( "Question 846461: Using the letters in \"PROPOSITIONS\"\r
\n" ); document.write( "\n" ); document.write( "a) How many of the distinguishable arrangements have the two Ps adjacent and the three Os adjacent?\r
\n" ); document.write( "\n" ); document.write( "b) How many of the distinguishable arrangements do not have the two Ps adjacent? \n" ); document.write( "

Algebra.Com's Answer #512078 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "         PROPOSITIONS\r\n" );
document.write( "it has 2 P's, 3 O's, 2 S's, 2 I's, 1 R, 1 T, 1 N\r\n" );
document.write( "

\n" ); document.write( "Using the letters in \"PROPOSITIONS\"\r
\n" ); document.write( "\n" ); document.write( "a) How many of the distinguishable arrangements have the two Ps adjacent and the three O's adjacent?
\n" ); document.write( "

\r\n" );
document.write( "That's the distinguishable arrangements of these 9 things\r\n" );
document.write( "{PP,OOO,R,S,S,I,I,T,N}\r\n" );
document.write( "\r\n" );
document.write( "The two S's are indistinguishable and the two I's are indistinguishable:\r\n" );
document.write( "\r\n" );
document.write( "That's  = 90720\r\n" );
document.write( "

\n" ); document.write( "b) How many of the distinguishable arrangements do not have the two Ps adjacent?
\n" ); document.write( "

\r\n" );
document.write( "First we find the number of distinguishable arrangements:\r\n" );
document.write( "\r\n" );
document.write( " = 9979200\r\n" );
document.write( "\r\n" );
document.write( "Next we find the number that have the 2 P's adjacent.\r\n" );
document.write( "\r\n" );
document.write( "That is the number of distinguishable arrangements of these 11 things\r\n" );
document.write( "\r\n" );
document.write( "{PP,O,O,O,S,S,I,I,R,T,N}\r\n" );
document.write( "\r\n" );
document.write( "The 3 O's, the 2 S's and the 2 I's are indistinguishable:\r\n" );
document.write( "\r\n" );
document.write( "That's  = 1663200\r\n" );
document.write( "\r\n" );
document.write( "So we subtract\r\n" );
document.write( "\r\n" );
document.write( "9979200 - 1663200 = 8316000\r\n" );
document.write( "\r\n" );
document.write( "Answer: 8316000\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );