Why can a product give a number of combinations?

Question

Mathematics

If we represent the product of apples by baskets, then we get the total number of apples in all baskets.
But if we need to calculate the number of combinations (factorial) of the location of 6 books on a shelf, we also need a product there. But how can one imagine this work in this case?
1 * 2 * 3 * 4 * 5 * 6 = 720 combinations of books (out of 6) can be placed on a shelf.
If we multiplied baskets by apples, we knew what we were multiplying by. Why multiply all intermediate numbers here? How can this be clearly described? (And why multiplication and not exponentiation?)

Answer 1

For the first book (out of six), there are six empty positions on the shelf. For the second - there are five for each of the six options for placing the first book (6 * 5). For the third - 4 for each of 30 (6*5*4). And so on, for the sixth, one position remains for each of the 720 (6*5*4*3*2) placement options for the first five books. Total 6*5*4*3*2*1 = 6! = 720 options.