You can use it any six days in the 31-day period between May 15 and June 14 (both included). Not June 15th though, as that would be a month and a day.

The actual month is only relevant for determining the exact number of days: 31 days as you start in May or if you would start during any other month with 31 days, 28 if you start at any point in February (unless it's a leap year, then 29), and 30 if you start your pass in the course of the months with 30 days.

Schelte, Thank you for clarifying. It can be a bit overwhelming. 

Understandably, glad I could help. I hope you enjoy your trip!