Number of ways of arranging $$8$$ identical books into $$4$$ identical shelves where any number of shelves may remain empty is equal to

Since both the books and the shelves are identical, this problem reduces to finding the number of partitions of 8 into at most 4 parts.

A partition of 8 into at most 4 parts means writing 8 as an unordered sum of at most 4 positive integers (where empty shelves correspond to parts of value 0).

Listing all partitions of 8 into at most 4 parts:

1 part:

8 = 8 → (8)

2 parts:

8 = 7+1, 6+2, 5+3, 4+4 → 4 partitions

3 parts:

8 = 6+1+1, 5+2+1, 4+3+1, 4+2+2, 3+3+2 → 5 partitions

4 parts:

8 = 5+1+1+1, 4+2+1+1, 3+3+1+1, 3+2+2+1, 2+2+2+2 → 5 partitions

Total = 1 + 4 + 5 + 5 = 15

Therefore, the correct answer is Option 4: 15.