Different Answer, Same Result

I did this and it worked:

db.companies.find({$or: [{founded_year: 2004, $or: [{category_code: "web"}, {category_code: "social"}]}, {founded_month: 10, $or: [{category_code: "social"}, {category_code: "web"}]}]}).count()

It longer that the proposed solution but I think is closer to the problem statement.
Is this correct?

To me, it looks perfectly fine, just repeats categories, but as you said, it follows a different way of thinking.