Quartominos

A Domino has two numbers, one at each end. A Triomino has three numbers, one in each corner of a triangle. A Quartomino has four numbers, one in each corner of a square.
  1. Find all 24 unique quartominos created by placing the numbers 0, 1 or 2 in each of the corners. A quartomino is unique if it cannot be rotated to make another.
  2. Place the 24 quartominos from part 1 into a 5x5 grid, duplicating only one, such that each quartomino matches its neighbors at both corners.
Source: Original.
Solution
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