Out of 13 objects, 4 are indistinguishable and the rest are distinct. The number of ways we can choose 4 objects out of 13 objects is

There are $256$ choices. Choose $k$ of the nine distinct objects for $k=0,1,2,3,4$, then fill the remaining places with indistinguishable objects, giving $1+9+36+84+126=256$. Leaving out the all-indistinguishable selection gives $255$.