JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Permutations and Combinations

Distributing 8 identical books into 4 identical shelves (empty shelves allowed) is equivalent to writing $8$ as a sum of at most $4$ positive parts — i.e. counting partitions of $8$ into at most $4$ parts.

Group them by the number of non-empty shelves used:

1 part: $8$ — 1 way.

2 parts: $7+1,6+2,5+3,4+4$ — 4 ways.

3 parts: $6+1+1,5+2+1,4+3+1,4+2+2,3+3+2$ — 5 ways.

4 parts: $5+1+1+1,4+2+1+1,3+3+1+1,3+2+2+1,2+2+2+2$ — 5 ways.

Total $=1+4+5+5=15$ ways.