Change at the Lemonade Stand

A lemonade stand charges $\$5$ a cup and opens with an empty cash box. $2n$ customers line up in uniformly random order: $n$ carry exact change (a single $\$5$ bill) and $n$ carry only a $\$10$ bill and will need $\$5$ back. Serving strictly in line order, what is the probability the stand never gets stuck unable to make change?