document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191176>Question 1191176</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3. In how many ways can 9 different books be arranged on a shelf so that <BR>\n" );
document.write( "(a) 3 of the books are always together, <BR>\n" );
document.write( "(b) 3 of the books are never all 3 together?\r<BR>\n" );
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document.write( "Show full solution, thank you! </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "We have 9 books which I'll call <BR>\n" );
document.write( "book a, book b, book c, ..., book h, book i<BR>\n" );
document.write( "Let's say we want books a through c to always stick together in any particular order.\r<BR>\n" );
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document.write( "We can pull books a,b,c out of the group on a temporary basis for now. Replace them with book j.<BR>\n" );
document.write( "The position of book j will represent books a,b,c in any order.<BR>\n" );
document.write( "The 9 books drop to 9-3 = 6 after taking books a,b,c out, but then the count bumps up to 6+1 = 7 after adding book j.\r<BR>\n" );
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document.write( "Arrange the 7 books and you should find there are 7! = 7*6*5*4*3*2*1 = 5040 permutations. The order matters. <BR>\n" );
document.write( "You can alternatively use the nPr formula with n = 7 and r = 7.\r<BR>\n" );
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document.write( "That 5040 describes sequences involving book j. Wherever you see book j, replace it with some permutation of a,b,c.<BR>\n" );
document.write( "For example, if we had the sequence<BR>\n" );
document.write( "<font color=blue>j</font>,d,e,f,g,h,i<BR>\n" );
document.write( "then it could represent any of the following 6 items<ul><li><font color=blue>a,b,c</font>,d,e,f,g,h,i</li><li><font color=blue>a,c,b</font>,d,e,f,g,h,i</li><li><font color=blue>b,a,c</font>,d,e,f,g,h,i</li><li><font color=blue>b,c,a</font>,d,e,f,g,h,i</li><li><font color=blue>c,a,b</font>,d,e,f,g,h,i</li><li><font color=blue>c,b,a</font>,d,e,f,g,h,i</li></ul>The 6 is due to the fact that 3! = 3*2*1 = 6<BR>\n" );
document.write( "There are 6 ways to arrange any group of 3 items where order matters.\r<BR>\n" );
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document.write( "As another example, the sequence here<BR>\n" );
document.write( "g,h,i,<font color=blue>j</font>,d,e,f<BR>\n" );
document.write( "could represent any of the following<ul><li>g,h,i,<font color=blue>a,b,c</font>,d,e,f</li><li>g,h,i,<font color=blue>a,c,b</font>,d,e,f</li><li>g,h,i,<font color=blue>b,a,c</font>,d,e,f</li><li>g,h,i,<font color=blue>b,c,a</font>,d,e,f</li><li>g,h,i,<font color=blue>c,a,b</font>,d,e,f</li><li>g,h,i,<font color=blue>c,b,a</font>,d,e,f</li></ul>Overall, there are 6*5040 = <font color=red>30,240</font> ways to arrange the 9 books such that 3 of them are always together in some fashion.\r<BR>\n" );
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document.write( "Answer: <font color=red>30,240</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "There are 9! = 9*8*7*6*5*4*3*2*1 = 362,880 different ways to arrange all of the books regardless if 3 particular books stick together or not.\r<BR>\n" );
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document.write( "We found earlier there are 30,240 ways to arrange the books so that 3 stick together.\r<BR>\n" );
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document.write( "Subtract those two values to get<BR>\n" );
document.write( "362,880 - 30,240 = <font color=red>332,640</font><BR>\n" );
document.write( "which represents the number of ways to have the 3 books separated in some fashion. Either they are all isolated from one another, or we only have 2 of them together (but not all 3 together).\r<BR>\n" );
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document.write( "This works because those 3 books are either always together, or split apart in some way. The two events are complementary of one another.\r<BR>\n" );
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document.write( "Answer: <font color=red>332,640</font>\r<BR>\n" );
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