Thanks for your question.
Assuming that you and your friend have entered a straightforward
pick-a-name-out-of-a-hat type of lottery (i.e. all 25 names are put in
the hat, and the first one picked out gets first prize, and the second
name gets second prize), then the odds can be figured like this:
1). To win $17,500, then two things have to happen: one of you wins
first place, and the other, second place.
The odds of winning first place are 2 out of 25 -- either you or your
friend can win.
The odds of the remaining friend (who didn't win first place) then
winning second place are 1 out of 24, since there are only 24 names
now left in the hat.
So the odds of both happening are:
2/25 x 1/24 = 0.00333333333 or
1/3 of 1% or
3.3 chances out of 1,000
2). To win $10,000 also requires that two things happen: one friend
wins first place, and the second friend does NOT win second place.
The odds of one of the friends winning first place are same as before
-- 2 out of 25.
The odds of the second friend NOT winning second place are 23 out of 24.
The odds of both happening are:
2/25 x 23/24 = 0.0766666667 or
between 7 and 8 (seven and two-thirds, actually) chances in 100.
I hope I've presented this clearly. But before rating this answer,
please let me know if you need any additional information. Just post
a Request for Clarification, and I'll be happy to assist you further.
No search strategy was used for this question, as I was able to answer
it based on my own understanding of numbers.