In how many ways can $n$ identical socks be arranged in a drawer with $k \geqslant 1$ compartments.
Each drawer is associated with a number from 1 to $k$. Each way of arranging the socks is represented using a transparent tube and $n+k-1$ ping-pong balls, of which $n$ are white and $k-1$ are orange. Indeed, if the tube starts with $a_{1}$ white balls, then we put $a_{1}$ socks in drawer 1. If the tube continues with an orange ball and $a_{2}$ white balls, then we put $a_{2}$ socks in drawer 2. Since there are $\binom{n+k-1}{k-1}$ ways to place the ping-pong balls in the tube (positions of the orange balls), there are as many ways to arrange the socks in the drawers.