Question 666126
There are 8 cities to be visited.

We need to choose 3,4,5,6,or 7 cities to visit.

So there are (8 choose 3) + (8 choose 4) + (8 choose 5) + (8 choose 6) + (8 choose 7) = 218 ways to visit 3,4,5,6,or 7 cities.