To solve the problem of finding the number of ways to arrange 12 distinct books into 3 distinct boxes with each box having at least one book, we use the inclusion-exclusion principle for surjective functions (since we need every box to contain at least one book).

Step 1: Total ways without restrictions

Each book has 3 choices (any box), so:
[3^{12} = 531441]

Step 2: Subtract ways where at least one box is empty

Choose 1 box to be empty (3 choices), then each book goes to the remaining 2 boxes:
[3 \times 2^{12} = 3 \times 4096 = 12288]

Step 3: Add back ways where two boxes are empty

Choose 2 boxes to be empty (C(3,2)=3 choices), then all books go to the remaining box:
[3 \times 1^{12} = 3]

Step 4: Apply inclusion-exclusion

[ \text{Valid ways} = 3^{12} - 3 \times 2^{12} + 3 \times 1^{12} ]
[ = 531441 - 12288 + 3 = 519156]

Answer: (\boxed{519156})

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