I think the analogy with the permutations of letters is making this problem more complicated than it needs to be.
Using the restriction that the number has at least one seven, you can first find the numbers that have exactly one $7$, then the numbers that have two $7$s, and then the number that has three $7$s and then add the results.
To find the number of numbers, think of choosing a digit for each spot: _ _ _
For one seven, you can fix a $7$ in a spot, say the first one, so the number looks like 7_ _ and note that for the other spots you can have any of the other 9 digits ($0,1,2,3,4,5,6,8,$ or $9$), so there are $9\cdot 9=81$ such numbers.
For numbers of the forms _ 7 _ and _ _ 7 the count is different because the first digit cannot be $0$, so there are $8\cdot 9=72$ possibilities for each.
Thus, in total there are $72+72+81=225$ three-digit positive integers with one seven as a digit.
Two sevens: For the form _77 there are 8 possibilities because the first spot cannot be 0, and for each of the forms 7_7 and _ _7 there are 9 possibilities, so in total there are $8+9+9=26$ three-digit positive integers with one seven as a digit.
Three sevens: There is only one, $777$.
So in total there are $225+26+1=252$ three-digit integers with a seven as a digit.
You’re not nuts! But what your insight is really revealing is that this question is ambiguous. If there is a team A and a team B, then the book is right. But if the teams are not to be distinguished, then you are right. So which is it? We don’t know. That’s why the question is ambiguous.
Still, in a real life situation, where 10 players on a basketball court decide to create two teams, I would say your answer is more appropriate.
Best Answer
I think you are almost right, except that you forgot to consider the empty collection of cans, which brings the total number to $240$. Another way to get to the same result is the following argument: a combination of cans is uniquely determined by the number of cans of each type: so you have $6$ possibilities for the Coca-Cola type (namely, you can pick $0$, $1$,..., or $5$ cans), and similarly $5$ possibilities for the Seven-Up and $8$ for the Mountain Dew. So in total you'll have $6\cdot 5\cdot 8=240$ possible combinations of cans.