A set of $4$ elements has $2^4$ subsets. These include the empty set, in which case you eat nothing. So the answer is $16$ if eating nothing is an option, or $15$ if eating nothing is not an option. The wording of the problem does not make it clear whether you can choose the empty set of foods.
The numbers are small enough that you could actually enumerate all the possibilities. You were well under way. To be sure of not missing any, you might proceed systematically. For brevity call the foods A, B, C, D.
We can have
ii) A or B or C or D;
iii) AB or AC or AD or BC or BD or CD;
iv) BCD or ACD or ABD or ABC;
Count: we get $16$.
To see the "formula" $2^4$, imagine you are in a cafeteria line and the A's, B's, C's, and D's are lined up in that order. You look at the A's and say yes or no. Then you do the same with the B's, the C's, the D's. The number of possible choices is the number of strings of length $4$ made up of the letters Y (for Yes) and/or N (for No). There are $2^4$ such strings. The option NNNN corresponds to going hungry.