Final answer:
The total number of ways a person can enter the gymnasium through one door and leave through a different door is 30.
Step-by-step explanation:
To calculate the number of ways a person can enter the gymnasium through one door and leave through a different door, we need to consider the number of options for entering and the number of options for leaving. Let's assume there are 6 doors in total. The person can choose any one of the 6 doors to enter and any one of the remaining 5 doors to leave. Therefore, the total number of ways is 6 * 5 = 30.