Chess with Maths

In how many ways two queens shall be placed into 8×8 chessboard such that they never get cancel by each other ( there are only two queens on board place , assume white plays first ).

I found the words "that they never get cancel by each other". It is hard to get... If you did not combine the definition of the second sentence "assume white plays first", it is hard to know the real definition of the word cancel...

There are two definition:

1. decide or announce that (a planned event) will not take place.

2. (of a factor or circumstance) neutralize or negate the force or effect of (another).

The first one says that, capturing a piece is included in the problem... Because, if you cancel the effects of the other piece, you must capture it while the ones who capture remain hence "they did not cancel each other"...

The second one says that, capturing a piece is not included in the problem... Because, if you capture the other, it will not exist...

Because most of us agreed that the second definition is what we are talking about... I guess, the second definition seems more practical...

Yes!

There are 42 ways the Qs can be placed with the white Q on a1. There are 42 ways if the white Q is on a4. If the white Q is on f4, there are 38. Logically, for all the edge squares there are 42. If on b7 there are 40, so a pattern is developing. If we divide the board into concentric tracks, the outer track contains squares all of which are 42. The next track inwards 40, the next track inwards, 38.

The final track is the centre four squares. Taking one at random, d4, there are 5x6 +6 = 36

So we have 4x36 + 12x38 + 20x40 + 28x42 = 2576

Double it and we have the answer.

5152

Well as per your question

I have used the word cancel because if I would write capture then definitely some will come up and will say capturing has has only one ways

Also capturing is the choice whether the queens would like to do capture or not

As per your concern I will edit question for less confusions

I think you can change the cancel to "in a line of sight" just to be clear and please tell me what queen color is being used in this puzzle

Its clear that two queens of same color cannot capture or cancel

It's clear they cannot capture. Whether they can cancel is anybody's guess, it depends what you intended to mean by the term. If you intended to mean just "capture" then two queens of the same colour satisfy your criterion wherever they are placed, so all such diagrams should be included in the count unless you explicitly say the queens are of opposite colour. 

If you consider queens of opposite colour, then the answer would depend on what you intended by the phrase "such that they never get cancel by each other". The inclusion of the word "never" seems to imply that the game continues. In that case, assuming "cancel" is synonymous with "capture", there are 0 diagrams where that is the case in all possible continuations (though it could be the case from any diagram if e.g. a draw has been agreed). 

Edit: I notice you have have clarified that "cancel" is to be taken as an approximate synonym of "capture" and removed the reference to White having the move. The question is probably about only the diagram and any game continuation is irrelevant, in which case

In how many ways can a queen of each colour be placed on an otherwise empty chessboard in such a way that neither queen occupies a square attacked by the other?

would have been clearer.

If the game has 2 queens you have to count for the both first and second that's the reason answer get doubled .

No. You may have to change your id. 

As you know chess board have coordinates so placing on that coordinates from any of the sides are totally different positions

Its clear that two queens of same color cannot capture or cancel

It's clear they cannot capture. Whether they can cancel is anybody's guess, it depends what you intended to mean by the term. If you intended to mean just "capture" then two queens of the same colour satisfy your criterion wherever they are placed, so all such diagrams should be included in the count unless you explicitly say the queens are of opposite colour. 

If you consider queens of opposite colour, then the answer would depend on what you intended by the phrase "such that they never get cancel by each other". The inclusion of the word "never" seems to imply that the game continues. In that case, assuming "cancel" is synonymous with "capture", there are 0 diagrams where that is the case in all possible continuations (though it could be the case from any diagram if e.g. a draw has been agreed). 

Edit: I notice you have have clarified that "cancel" is to be taken as an approximate synonym of "capture" and removed the reference to White having the move. The question is probably about only the diagram and any game continuation is irrelevant, in which case

In how many ways can a queen of each colour be placed on an otherwise empty chessboard in such a way that neither queen occupies a square attacked by the other?

would have been clearer.

If the game has 2 queens you have to count for the both first and second that's the reason answer get doubled .

No. You may have to change your id. 

As you know chess board have coordinates so placing on that coordinates from any of the sides are totally different positions

Its clear that two queens of same color cannot capture or cancel

It's clear they cannot capture. Whether they can cancel is anybody's guess, it depends what you intended to mean by the term. If you intended to mean just "capture" then two queens of the same colour satisfy your criterion wherever they are placed, so all such diagrams should be included in the count unless you explicitly say the queens are of opposite colour. 

If you consider queens of opposite colour, then the answer would depend on what you intended by the phrase "such that they never get cancel by each other". The inclusion of the word "never" seems to imply that the game continues. In that case, assuming "cancel" is synonymous with "capture", there are 0 diagrams where that is the case in all possible continuations (though it could be the case from any diagram if e.g. a draw has been agreed). 

Edit: I notice you have have clarified that "cancel" is to be taken as an approximate synonym of "capture" and removed the reference to White having the move. The question is probably about only the diagram and any game continuation is irrelevant, in which case

In how many ways can a queen of each colour be placed on an otherwise empty chessboard in such a way that neither queen occupies a square attacked by the other?

would have been clearer.

If the game has 2 queens you have to count for the both first and second that's the reason answer get doubled .

No. You may have to change your id. 

As you know chess board have coordinates so placing on that coordinates from any of the sides are totally different positions