I guess what sounds authoritative is that I'm using jargon while trying to explain the jargon at the same time, as if I knew what I was talking about. I've since edited my comment. I know it's stars and bars, but I keep messing up the count. Anyway, it's a finite amount, it's not huge, and your count is wrong too by a sister comment. :-)
Not true. The problem is in one-to-one correspondence with the non-negative solutions to
a + b + c + d + e = 25
which is in one-to-one correspondence with the positive solutions to
a + b + c + d + e = 30
which is a simple counting problem---(29 choose 4) = 23751.
In my opinion, you should really try to prove your math to yourself before answering with such an authoritative tone.