Question 163939
I'll take a stab at it
Here's 6 teams, 5 students each:
team 1: a b c d e
team 2: a b c d e
team 3: a b c d e 
team 4: a b c d e 
team 5: a b c d e 
team 6: a b c d e
Assume 1a gets the gold. How many ways can the other teams
get the Silver?
There are 25 students in the remaining 5 teams, so there are
25 ways the Silver can be given
If 1b gets the Gold, also 25 ways to give Silver, etc
So, if someone from team 1 gets Gold, there are
{{{5*25 = 125}}} ways to give Silver 
If someone from team 2 gets Gold, also {{{125}}} ways
So, for all 6 teams:
{{{6*125 = 750}}}ways to give Silver
Now 1 team has Gold and 1 team has Silver
There are 20 students in the remaining 4 teams
For each of the {{{750}}} combinations of Gold and Silver,
there are 20 ways to give Bronze.
{{{750*20 = 15000}}}
There are 15000 different winning arrangements
This seems pretty large. Check my reasoning-maybe it's off