What is the probability that any two students had atleast one county in common?

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There are 20 students in class and each student was asked to give a chart for 10 different counties in the state of Texas. Here in the state of Texas, there are 254 counties

So what I am doing here is basically using combinations to find the counties such that there are not any common for that I am going with.

C(254,10) * C(244,10) * C(234,10) ....so on till * C(64,10) / C(254,10)^20

is this the approach correct or any other suggestions?