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

Solution

Maths sir · 3 steps

Books identical and shelves identical: only the sizes of the piles matter, so list partitions of 8.

1. What counts. The books are identical, so only how many sit on each shelf matters. The shelves are identical too, so the order of those numbers does not matter. Each way is a partition of 8 into at most 4 parts.
2. List by shelves used. Write each partition with its parts in decreasing order so no case is counted twice. Then group the partitions by how many shelves are non-empty.

   Non-empty shelves	Partitions of 8	Count
   1	8	1
   2	7+1, 6+2, 5+3, 4+4	4
   3	6+1+1, 5+2+1, 4+3+1, 4+2+2, 3+3+2	5
   4	5+1+1+1, 4+2+1+1, 3+3+1+1, 3+2+2+1, 2+2+2+2	5

3. So. Total = 1 + 4 + 5 + 5 = 15, which is option (d).

The idea to keep. Identical objects into identical boxes has no neat formula: list the partitions in decreasing order, grouped by the number of non-empty boxes.

Answer from the NTA final answer key. Solution by PrepWiser.