"I apply for a free trip ticket."
Translation:공짜 여행권을 신청해요.
권 - entry ticket (in the sense of a pass, coupon - used mainly for leisure places) 표 - ticket (for general usage esp. in transports)
In this example, 권 is used probably to mean the 'free ticket' is some form of coupon.
Also worth noting that Koreans have the expression 자유이용권 for free pass - a reason for DLG to use 권 in this example perhaps.
But using "표" instead of "권" here is not incorrect. Just avoid using both together.
Note: there is also the konglish word "티켓" for ticket.