Algebra.Com's Answer #674583 by ikleyn(52814)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Let  be the set of all sitting arrangements of A, B, C and D in four seats.\r\n" );
document.write( "As everybody knows, consists of 4*3*2*1 = 24 permutations.\r\n" );
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document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A is sitting in the seat #1.\r\n" );
document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where B is sitting in the seat #2.\r\n" );
document.write( "Let  be . . . where C is sitting in the seat #3.\r\n" );
document.write( "Let  be . . . where D is sitting in the seat #4.\r\n" );
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document.write( "Then is, oviously, the union of subsets P + + + + , (1)\r\n" );
document.write( "where P is our \"most desired\" subset of those sitting arrangements that are under the problem's question.\r\n" );
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document.write( "It is clear that the cardinality of each subset , , and is 6 = 3*2*1.\r\n" );
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document.write( "Therefore, the first desire is to write 24 = |P| + 6 + 6 + 6 + 6, following (1).\r\n" );
document.write( "But it would not to be correct.\r\n" );
document.write( "Why? - Because the subsets , , and have non-empty intersections that we counted twice.\r\n" );
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document.write( "Did you just get understanding on how to solve the problem and how to proceed?\r\n" );
document.write( "If not, then follow me.\r\n" );
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document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A and B are sitting in the seats #1 and #2, respectively.\r\n" );
document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A and C are sitting in the seats #1 and #3, respectively.\r\n" );
document.write( "Let  ,  ,  and  be the other four similar subsets in .\r\n" );
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document.write( "Then we can make a correction to formula (1) by writing\r\n" );
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document.write( " = P + + + + - - - - - - (2)\r\n" );
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document.write( "in the effort to account for double intersections.\r\n" );
document.write( "But be aware !! This formula is still not exactly correct.\r\n" );
document.write( "Why? - Because the sets , , , , and have triple intersections, which we distracted twice in the formula (2).\r\n" );
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document.write( "Did you just get understanding on how to solve the problem and how to proceed?\r\n" );
document.write( "If not, then follow me again.\r\n" );
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document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A, B and C are sitting in the seats #1, #2 and #3, respectively.\r\n" );
document.write( "Let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A, B and D are sitting in the seats #1, #2 and #4, respectively.\r\n" );
document.write( "Let  and  be the other two similar subsets in .\r\n" );
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document.write( "At last, let  be the subset of all sitting arrangements of A, B, C and D in four seats, where A, B, C and D are sitting in the seats #1, #2, #3 and #4, respectively.\r\n" );
document.write( " (As you understand, the set consists of only ONE element, the identical permitation).\r\n" );
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document.write( "Now, finally, we can write the formula which is ABSOLUTELY true! :\r\n" );
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document.write( " = P + + + + - - - - - - + + + + - . (3)\r\n" );
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document.write( "I just mentioned above that the cardinality of the 1-letter subsets , , and is 6 = 3*2*1.\r\n" );
document.write( "The cardinality of the 2-letter subsets , , . . . , is 2 = 2*1. Obviously.\r\n" );
document.write( "The cardinality of the 3-letter subsets , , . . . , is 1. Obviously.\r\n" );
document.write( "The cardinality of the 4-letter subsets is 1. Obviously.\r\n" );
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document.write( "Therefore, following to the formula (3), we can write its \"cardinality\" analogue\r\n" );
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document.write( "24 = |P| + (6 + 6 + 6 + 6) - (2 + 2 + 2 + 2 + 2 + 2) + (1 + 1 + 1 + 1) - 1 = |P| + 4*6 - 2*6 + 4 - 1 = |P| + 24 - 12 + 4 - 1 = |P| + 15.\r\n" );
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document.write( " 4 times 6 times 4 times\r\n" );
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document.write( "Again, our equation is\r\n" );
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document.write( "24 = |P| + 15.\r\n" );
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document.write( "From here, the cardinality of the set P is |P| = 24 - 15 = 9.\r\n" );
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document.write( "Answer. How many ways are there if Alex does not sit in the first place, Bob does not sit in the second place, \r\n" );
document.write( " Charles does not sit in the third place, and Dan does not sit in the fourth place? - 9 ways.\r\n" );
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document.write( "I understand that this solution may seem to be complicated for a novice.\r
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document.write( "Yes, it would be good to be prepared for such endeavors.\r
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document.write( "There are two lessons in this site closely related to this theme that I can recommend you:\r
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document.write( " - Counting elements in sub-sets of a given finite set\r
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document.write( " - Advanced problems on counting elements in sub-sets of a given finite set\r
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document.write( "Also, you have these two free of charge online textbooks in ALGEBRA in this site\r
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document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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document.write( " - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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document.write( "The referred lessons are the parts of these online textbooks under the topic \"Miscellaneous word problems\".\r
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document.write( "From my side, it was a pleasure to me to work on this solution.
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document.write( "Thanks for posting this challenging problem.\r
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