Probability household cars problem A survey consists of recording, the number of cars presently owned by a household among six major manufacturers. Order does not matter. i. Suppose that no household has more than four cars. In how many different ways can the survey sheet be filled out? I was thinking maybe it would just be 4! + 6! but I have a feeling that is wrong, I'm not sure if that would account for multiple cars being from the same brand

Let $x_1$ be the number of GMs, let $x_2$ the number of Fords, and so on up to $x_6$ being the number of Nissans. We want to find the number of solutions of $$x_1+x_2+\cdots+x_6\le 4$$ in non-negative integers. This is the same as the number of solutions of $$x_1+x_2+\cdots+x_6+x_7= 4$$ (the variable $x_7$ counts the number of empty slots in the four-car garage).

Now we have a standard _Stars and Bars_ problem (please see Wikipedia). The number of solutions is $\binom{4+7-1}{7-1}$, that is, $\binom{10}{6}$, or equivalently $\binom{10}{4}$.