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Find the number of ways in which 2 identical kings can be placed on an n x m chessboard so that the kings are not in adjacent squares.

Find the number of ways in which 2 identical kings can be placed on an n x m chessboard so that the kings are not in adjacent squares.

Grade:10

4 Answers

pankaj kumar raman
51 Points
9 years ago

first king can be placed in n*m places

1)one king at corner ,other king will be at = 4  *  [(n*m)-4] places    = 4mn-16

                                                              ↑          ↑

                                                     4 corners   [m*n places - 4 places around] 

 

2)one king at edge, other will be at= [2(n-2)+2(m-2)]  *  (n*m-5) = 2mn²-10n+2m²n-10m-8mn+40

                                                         ↑                         ↑

                                       number of box at edge      m*n places except 5 places around

 

3)one king at free from corners and edges, other will be at

                      = [(n-2)*(m-2)]     *        (n*m-8)      =   n²m²-2n²m-2nm²-4nm+16n+16m-32

                                ↑                                 ↑  

  number of boxes away from ends           n*m places except 8 places around

 

 

sum of all probablities will be= (nm-4)²+6(m+n-4)

kushal satya
37 Points
9 years ago

A

 

 

 

 

 

 

 

 

B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D

 

 

 

 

 

 

 

 

C

 

 

           
 

 LET THE ABOVE BE THE CHESS BOARD.

WHEN ONE OF THE KING IS PLACED AT A,B,C,D THEN NO OF POSITIONS AT WHICH THE OTHER KING COULD BE PLACED IS mn-4 (three adjacent and 1 the position of the first king).

hence no of ways in this case = 4(mn-4)

no of ways when first king is placed between a and b or b and cand so on in the same row or column then no of ways=2(m+n-4)(mn-6)

in the last case when the king is placed elsewhere on the board,

no of ways =2(mn-m-n)(mn-9)

add all the ways to get the net no of ways.

     
 
 
     
 
     

 

APURV GOEL
39 Points
9 years ago

sorry this isnt the ans.

pankaj kumar raman
51 Points
9 years ago

then tell me the right solution

as it was disapproved

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