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(Building intuition)
Start with 2 items: A, B. How many ways can you arrange them? List them: AB, BA. That's 2 ways.
Now 3 items: A, B, C. For the first spot, you have 3 choices. For the second, 2 remain. For the third, 1 is left. Total: 3 × 2 × 1 = 6 ways.
The pattern: multiply the number of choices at each step. This product appears everywhere in permutations.
