let reduce this into small size problem
let there are 4 people named them as A,B,C and D, we want to divide them as size of 2
Formula is C(4,2) = 6
what are those ?
i) ( A , B )
ii) ( A , C )
iii) ( A , D )
iv) ( B , C )
v) ( B , D )
vi) ( C , D )
1) As per First question, Teams are Numbered ===> the first team can be any one of the choice, after choosing that team there is no choice for choosing
like if i choose team A = (A,B) then i should choose team B = (C,D)
∴ Total Possibilities = 6 = C(4,2)
2) in the second question, Teams are Not Numbered, means
if you are saying choose (A,B) this is equivalent to choose (C,D) as team
if you are saying choose (A,C) this is equivalent to choose (B,D) as team
if you are saying choose (A,D) this is equivalent to choose (B,C) as team
∴ Total Possibilities = 3 = $\frac{6}{2}$ = $\frac{ C(4,2)}{2}$
For your question
A) C(10,5)
B) $\frac{ C(10,5)}{2}$