Question 1210171
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Find the number of ways of placing three As, three Bs, and three Cs in a 3x3 grid, 
so that every square contains one letter, and each diagonal contains one A, one B, and one C.
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<pre>
In this problem, three As are indistinguishable, as well as three Bs and three Cs.
Therefore, the subject of consideration are not permutations - the subject of
consideration are distinguishable arrangements.


Looking at the diagonal cells (1,1), (2,2), (3,3), three different letters A, B and C can be 
placed there in 6 different distinguishable ways.


(1)  Let assume that the letter in upper left corner (1,0) is A
         and that the letter in central cell (2,2) is B.
     
     Then the letter in the cell (3,3) is C inevitably.

     In the cell (3,1) we have choice of two letters A or C
     Then in the cell (1,3) we must place C or A oppositely, with no choice.

     The remaining cells (1,2), (2,1), (2,3) and (3,2) we can fill with remaining letters, 
     with no constraints. 

     The remaining letters are A, two Bs and C.

     This 4 letters can be placed in the remaining 4 cells by {{{(4*3*2)/2)}}} = 12  different distinguishable ways .

     Thus, starting from A and B in (1,1) and (2,2), we have  2*12 = 24 choices for placing the other letters.



(2)  Let assume that the letter in upper left corner (1,0) is B
         and that the letter in central cell (2,2) is A.

     Then, reasoning by the same way, we will have 24 different distinguishable choices 
     in placing all other letter.



(3)  Let assume that the letter in upper left corner (1,0) is A
         and that the letter in central cell (2,2) is C.

     Then, reasoning by the same way, we will have 24 different distinguishable choices 
     in placing all other letter.



(4)  Let assume that the letter in upper left corner (1,0) is C
         and that the letter in central cell (2,2) is A.

     Then, reasoning by the same way, we will have 24 different distinguishable choices 
     in placing all other letter.



(5)  Let assume that the letter in upper left corner (1,0) is B
         and that the letter in central cell (2,2) is c.

     Then, reasoning by the same way, we will have 24 different distinguishable choices 
     in placing all other letter.


(6)  Let assume that the letter in upper left corner (1,0) is C
         and that the letter in central cell (2,2) is B.

     Then, reasoning by the same way, we will have 24 different distinguishable choices 
     in placing all other letter.



Thus we analyzed all 6 possible placements of letters on the major diagonal,
and we saw that each such placing creates 24 different placing for the rest of letters.


Therefore, the total number of all possible distinguishable placing (arrangements) 
of the letters in this problem is 

    6 * 24 = 144.      <<<---===  <U>ANSWER</U>
</pre>

Solved.