Partially correct series calculations?

I know how to get the chances of drawing n numbered balls out of a hat in the correct order (n!).

How would I work out the chances of drawing n balls with at least q in the right order?

(for example drawing 20 numbered balls out of a hat at random and ending up with at least 10 in the right order)
Update: There can be an infinite number of balls in the hat. What I mean by "in the right order" is not that number 4 immediately follows number 3, its that at least ten of the balls are in "sequence" (eg 2, 5, 67, 89 etc).
Update 2: Come to think of it the "infinite balls" bit might be a little extreme. Take it as twenty balls in the hat if its easier!
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