No Change At The Movies

Ten students want to buy a ticket for a movie. The ticket costs $10. Five of them have a $10 note, and the other five have a $20 note. They all line up in a random order, and each uses their note to buy the ticket. Of course, those with $20 expect a change, and unfortunately, you don't have any money to start with. You can use the $10 you receive to make a change for the $20, though. What is the probability that everyone will receive their ticket and change as desired?

Your answer can be represented as \(\dfrac{a}{b}\) where \(a,b\) are non-negative integers that are relatively coprime. Enter your answer as \(a+b\).

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