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

Step-by-step solution

We have 4 identical shelves in which any number of shelves can be empty. So, we will put 3 , 2 , 1 , 0 shelves empty one by one. Also, the non-empty shelves won't require internal arrangement as all the shelves are identical. 3 Shelf empty: 8 , 0 , 0 , 0 → 1 way 2 shelf empty: ( 7 , 1 , 0 , 0 ) ( 6 , 2 , 0 , 0 ) ( 5 , 3 , 0 , 0 ) ( 4 , 4 , 0 , 0 ) → 4 ways 1 shelf empty: ( 6 , 1 , 1 , 0 ) ( 3 , 3 , 2 , 0 ) ( 5 , 2 , 1 , 0 ) ( 4 , 2 , 2 , 0 ) ( 4 , 3 , 1 , 0 ) → 5 ways 0 shelf empty: ( 1 , 2 , 3 , 2 ) ( 5 , 1 , 1 ,