For your question, I think you want (8 choose 1)*(8 choose 0)*(24 choose 6)/(40 choose 7)
Let me englishify this for you: the first term is how many ways we can pick one mountain from 8, next is how many ways to get 0 swamps from 8 (slightly tricky concept, but the answer is just 1 by definition*), then how many ways can the rest of the deck be ordered (huge number here), divided by how many ways can the whole deck be ordered (even bigger number).
I've found the time to check out that link now.
Now that I roughly understand that example, I also understand that it doesn't cover the whole issue. This is a single fail case where we have 1 Mountain and 0 Swamps - there are more: 0/0, 2/0, 3/0 etc. I would expect each of these events to have a (slightly) different probability - right?
How do I turn "8 choose 1" into a number? Your guide says it's a binomial coefficient.
Is there a point to simplifying the problem? I mean, we could think of a bowl with 40 balls, colored black, white or red. How many different groups of 7 can you pick? Early math classes cover those things with only 2 colors. How do we get into the 3rd "dimension"?