Choosing 3 teams out of 10 when order of the teams matters - \(P^3_{10}=\frac{10!}{7!}\);

Or: choosing which 3 teams out of 10 will get the medals - \(C^3_{10}\) and arranging them - \(3!\), so total - \(C^3_{10}*3!=\frac{10!}{7!}\);

Or:
1-2-3-4-5-6-7-8-9-10 (teams);
G-S-B-N-N-N-N-N-N-N (GSB - medals, N - no medal);

Permutation of 10 letters out of which 7 N's are identical is \(\frac{10!}{7!}\) (so you'll get \(\frac{10!}{7!}\) different ways of assigning the medals to the teams).

Answer: A.

Hope it's clear.
_________________

its choosing 3 teams from 10 teams when the order matters
so its 10P3.
the ans is A

I thought the ten choose three formula was (n choose k)= n!/(k!(n-k)!)

Am I over thinking this...

Agreed, A. I used slot method as order matters

Since the order matters (i.e., who gets gold, who gets silver and who gets bronze matters), the number of ways that 3 teams can be chosen from 10 teams is:

10P3 = 10 x 9 x 8 = (10 x 9 x 8 x 7!)/7! = 10!/7!

Answer: A
_________________