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.