How many ways are there to place $8$ indistinguishable balls into four distinguishable bins?

Let the number of balls in four distinguishable bins be $x_1,x_2,x_3,x_4$.

Then,

$x_1+x_2+x_3+x_4=8$

Here, balls are indistinguishable and bins are distinguishable.

So, the number of non-negative integral solutions is

${}^{n+r-1}C_{r-1}$

Here, $n=8$ and $r=4$.

So,

${}^{8+4-1}C_{4-1}={}^{11}C_{3}$

$=\dfrac{11 \times 10 \times 9}{3 \times 2 \times 1}$

$=165$