document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1131553>Question 1131553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In how many ways can 6 people be seated in a row if\r<BR>\n" );
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document.write( "(a) Persons A and B must sit next to each other?\r<BR>\n" );
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document.write( "(b) There are 3 men and 3 women, no 2 men and 2 women can sit next to each other? </SPAN>\n" );
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document.write( "(a) A and B must sit together.<br><BR>\n" );
document.write( "(1) Consider A and B as a \"single person\".  Then there are 5 \"people\" to be seated; this can be done in 5! = 120 ways.<BR>\n" );
document.write( "(2) In the group of A and B, the two can be seated in either of 2 different orders.<br><BR>\n" );
document.write( "ANSWER: 120*2 = 240 ways.<br><BR>\n" );
document.write( "(b) 3 men and 3 women; no 2 men and no 2 women can sit next to each other (i.e., the seating must alternate men and women).<br><BR>\n" );
document.write( "Consider seating the people one at a time, from the beginning of the row.<br><BR>\n" );
document.write( "(1) Any of the 6 people can be seated first: 6 choices<BR>\n" );
document.write( "(2) Without loss of generality, suppose the first person seated was a woman.  Then the second person seated must be one of the 3 men: 3 choices<BR>\n" );
document.write( "(3) The third person seated must be one of the remaining 2 women: 2 choices<BR>\n" );
document.write( "(4) The fourth person must be another man: 2 choices<BR>\n" );
document.write( "(5) The fifth person must be the 3rd woman: 1 choice<BR>\n" );
document.write( "(6) The sixth person must be the 3rd man: 1 choice<br><BR>\n" );
document.write( "The total number of ways to seat the 6 people is the product of the numbers of choices for each seat.<br><BR>\n" );
document.write( "ANSWER: 6*3*2*2*1*1 = 72 ways </SPAN>\n" );
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