Problem 7

Like many university lecturers, Dr Abiatha Grimecrooner receives a large number of emails to deal with each day. In fact, every day he receives 30 emails, all of which arrive before 5pm and all of which need replying to before he leaves work at 5:30pm.

Like many university lecturers, Abiatha prefers teaching his students and doing his research rather than replying to emails. Instead of blocking out time to deal with his over-flowing inbox, he replies to emails at various points throughout the day, operating a 'most-recent-in most-recent--out' policy, starting with the most recently received email and working back chronologically until he has either replied to all of the emails or has rushed off to do something else.

At 3pm, Abiatha replies to the 28th email that he has received that day. He always replies to all of his emails by the time he goes home at 5:30pm. How many possibilites are there for the ordered list of emails that Abiatha replies to between 4pm and 5:30pm?

For example, if you think the only possibility is that Abiatha replies to email 30 then to email 29, then there is one list (30,29) and you should write 1. Note that if you think it's possible that there are no emails to reply to then this counts as an ordered list (the empty list).