The best way to do these is to learn the multi-selection formula.

The number of ways to putidentical items intoNdifferent cells is .K

That is proven using the "stars & bars" as you suggested.

In your problem the identical items are the choices and the different cells are the varieties of doughnuts. But we have the requirement that no cell be empty. So put one of each variety in the box. Now count the number of ways to make an addition seven choices: .