NO of ways in which you can go from one corner of the chess board to the opposite corner of the chess board without retracing?
For going from one corner to another corner you have to make 8 right moves and 8 up moves or 8 down moves and 8 right moves.
Retracing here means we cant trace back any part of the path already covered.like if we have made 3 right moves and 2 up moves ,we cant gp down becuase then we are going back to same path.
So now we have to make 7 right moves and 7 up moves.The no of different combination will come ehen we go up when we go right .
Lets denote right move by R,up move up U.
So one way can be
another can be
By this time I think you must have guessed what we need to find out.We have to find distinct arrangements of 7R,7U,there is direct answer for this
14! divided by 7! * 7!
Bonus:Do you know the proof of arrangements of m things in which p are of one type,q are of one type,r of one type.