【Answer】 40
【Question Type】Combination: Maximum Construction
【Analysis】(1) Let the 10 cities be represented as 10 points $A_{1}, A_{2}, \cdots, A_{10}$, and the highways between two cities as line segments connecting two points, then at least two line segments connect any three points.
(2) First, connect all 10 points in pairs, resulting in a total of $C_{10}^{2}=45$ line segments (students who are not familiar with combination formulas can think about how to get 45 line segments).
(3) Now consider the maximum number of line segments that can be removed?
First, arbitrarily remove one line segment, let's say $A_{1} A_{2}$. Then the line segments $A_{1} A_{i}$ or $A_{2} A_{j}(i=3,4, \cdots, 10 ; j=3,4, \cdots, 10)$ cannot be removed, otherwise, there will be only one line segment among the three points $A_{1}, A_{2}, A_{i}$ or $A_{1}, A_{2}, A_{j}$. Therefore, only one line segment can be removed from the connections between $A_{3}, A_{4}, \cdots, A_{10}$, let's say $A_{3} A_{4}$; similarly, we can remove $A_{5} A_{6}, A_{7} A_{8}, A_{9} A_{10}$, so a maximum of 5 line segments can be removed.
(4) Therefore, at least $45-5=40$ line segments must be connected, meaning at least 40 highways must be opened.
10. Mom decides to take Xiao Hua on a car trip to 10 cities during the holiday. After checking the map, Xiao Hua is surprised to find: among these 10 cities, any three cities are either all connected by expressways, or only two of the cities are not connected by expressways. Therefore, there are at least $\qquad$ expressways between these 10 cities. (Note: There is at most one expressway between any two cities)