enya is standing in the centre square of a 9 by 9 grid. she travels outward one square at a time, moving diagonally or along a row of column. how many difrent paths can enya follow to the perimter?

Hello student,
Please find the answer to your question below
she has 320 unique paths to a perimeter square.
Standing on the center square, she has 8 possible squares she can move to to be in what one might call Ring 1 (relative to the single center square).
Those 8 squares are not identical in the number of Ring 2 squares they touch. The corner squares each touch 5 Ring 2 squares for a total of 20 unique paths. The squares between the corners (4 squares in this ring) each touch 3 Ring 2 squares for a total of 12 more unique paths: 20 + 12 = 32. So 40 so far.
The same thing happens in each ring as one moves outward. 4 corner squares with 5 choices, and however many (3, 5) in between with 3 choices each. They total 320.