Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may…

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: $\left(8,0,0,0\right)\to 1$ way

$2$ shelf empty: $\left.\begin{array}{c}(7,1,0,0)\\ (6,2,0,0)\\ (5,3,0,0)\\ (4,4,0,0)\end{array}\right]\to 4$ ways

$1$ shelf empty: $\left.\begin{array}{ll}(6,1,1,0)& (3,3,2,0)\\ (5,2,1,0)& (4,2,2,0)\\ (4,3,1,0)& \end{array}\right]\to 5$ways

$0$ shelf empty: $\left.\begin{array}{lr}(1,2,3,2)& (5,1,1,1)\\ (2,2,2,2)& \\ (3,3,1,1)& \\ (4,2,1,1)& \end{array}\right]\to 5$ ways

Total $=15$ ways