document.write( "Question 1181974: Ron is having a dinner party. He invites 7 out of his 10 friends. In how many ways can he choose his guests if two of them, Al and Sue, must attend together or not at all? \n" ); document.write( "
Algebra.Com's Answer #811937 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "Ron is having a dinner party. He invites 7 out of his 10 friends.
\n" ); document.write( "In how many ways can he choose his guests if two of them, Al and Sue, must attend together or not at all?
\n" ); document.write( "~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "As you understand, the answer is the sum of two quantities.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One quantity is the number of the ways, when both Al and Sue attend the party.\r\n" );
document.write( "\r\n" );
document.write( "This quantity is equal to  \"C%5B10-2%5D%5E%287-2%29\" = \"C%5B8%5D%5E5\" = \"%288%2A7%2A6%29%2F%281%2A2%2A3%29\" = 8*7 = 56\r\n" );
document.write( "\r\n" );
document.write( "     (because you simply select  5  guests from the  10-2 = 8  friends to add them to Al and Sue).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The other quantity is the number of the ways, that exculde both Al and Sue from the attendees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This quantity is  \"C%5B10-2%5D%5E7\" = \"C%5B8%5D%5E7\" = 8.\r\n" );
document.write( "\r\n" );
document.write( "    (because this time you select 7 friends not from 10 persons, but from 10-2 = 8 persons, only).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The final  ANSWER  is  56 + 8 = 64 ways.\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved, answered and carefully explained.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This problem is on COMBINATIONS.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On Combinations,  see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Introduction to Combinations\r
\n" ); document.write( "\n" ); document.write( "    - PROOF of the formula on the number of Combinations\r
\n" ); document.write( "\n" ); document.write( "    - Problems on Combinations\r
\n" ); document.write( "\n" ); document.write( "    - Problems on Combinations with restrictions \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( "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( "\n" ); document.write( " \n" ); document.write( "
\n" );