I think 권 and 표 are mostly equivalent. Perhaps one is a word from chinese and one is korean

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.

The hint for free does not match any of the possible answers. Android

공짜 - no (공=0) charge (짜 = cost) = free.

A synonym of 무료.