document.write( "Question 1210171: Find the number of ways of placing three As, three Bs, and three Cs in a 3 \times 3 grid, so that every square contains one letter, and each diagonal contains one A, one B, and one C. \n" ); document.write( "
Algebra.Com's Answer #851492 by ikleyn(52778)\"\" \"About 
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\n" ); document.write( "Find the number of ways of placing three As, three Bs, and three Cs in a 3x3 grid,
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document.write( "In this problem, three As are indistinguishable, as well as three Bs and three Cs.\r\n" );
document.write( "Therefore, the subject of consideration are not permutations - the subject of\r\n" );
document.write( "consideration are distinguishable arrangements.\r\n" );
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document.write( "Looking at the diagonal cells (1,1), (2,2), (3,3), three different letters A, B and C can be \r\n" );
document.write( "placed there in 6 different distinguishable ways.\r\n" );
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document.write( "(1)  Let assume that the letter in upper left corner (1,0) is A\r\n" );
document.write( "         and that the letter in central cell (2,2) is B.\r\n" );
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document.write( "     Then the letter in the cell (3,3) is C inevitably.\r\n" );
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document.write( "     In the cell (3,1) we have choice of two letters A or C\r\n" );
document.write( "     Then in the cell (1,3) we must place C or A oppositely, with no choice.\r\n" );
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document.write( "     The remaining cells (1,2), (2,1), (2,3) and (3,2) we can fill with remaining letters, \r\n" );
document.write( "     with no constraints. \r\n" );
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document.write( "     The remaining letters are A, two Bs and C.\r\n" );
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document.write( "     This 4 letters can be placed in the remaining 4 cells by \"%284%2A3%2A2%29%2F2%29\" = 12  different distinguishable ways .\r\n" );
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document.write( "     Thus, starting from A and B in (1,1) and (2,2), we have  2*12 = 24 choices for placing the other letters.\r\n" );
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document.write( "(2)  Let assume that the letter in upper left corner (1,0) is B\r\n" );
document.write( "         and that the letter in central cell (2,2) is A.\r\n" );
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document.write( "     Then, reasoning by the same way, we will have 24 different distinguishable choices \r\n" );
document.write( "     in placing all other letter.\r\n" );
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document.write( "(3)  Let assume that the letter in upper left corner (1,0) is A\r\n" );
document.write( "         and that the letter in central cell (2,2) is C.\r\n" );
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document.write( "     Then, reasoning by the same way, we will have 24 different distinguishable choices \r\n" );
document.write( "     in placing all other letter.\r\n" );
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document.write( "(4)  Let assume that the letter in upper left corner (1,0) is C\r\n" );
document.write( "         and that the letter in central cell (2,2) is A.\r\n" );
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document.write( "     Then, reasoning by the same way, we will have 24 different distinguishable choices \r\n" );
document.write( "     in placing all other letter.\r\n" );
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document.write( "(5)  Let assume that the letter in upper left corner (1,0) is B\r\n" );
document.write( "         and that the letter in central cell (2,2) is c.\r\n" );
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document.write( "     Then, reasoning by the same way, we will have 24 different distinguishable choices \r\n" );
document.write( "     in placing all other letter.\r\n" );
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document.write( "(6)  Let assume that the letter in upper left corner (1,0) is C\r\n" );
document.write( "         and that the letter in central cell (2,2) is B.\r\n" );
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document.write( "     Then, reasoning by the same way, we will have 24 different distinguishable choices \r\n" );
document.write( "     in placing all other letter.\r\n" );
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document.write( "Thus we analyzed all 6 possible placements of letters on the major diagonal,\r\n" );
document.write( "and we saw that each such placing creates 24 different placing for the rest of letters.\r\n" );
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document.write( "Therefore, the total number of all possible distinguishable placing (arrangements) \r\n" );
document.write( "of the letters in this problem is \r\n" );
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document.write( "    6 * 24 = 144.      <<<---===  ANSWER\r\n" );
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