There is an ambiguity in the question in that it is not clear whether or not the order of the five groups matters. The given answer tells us which was intended.
Two of the responders have given answers that are correct if the order of groups matters. This means choosing 2 people for group A, 2 for group B, and so forth through group E.
Miryana and sat have both implied that this is 113,400 and they found two valid ways to do it. One way is
(10 choose 2)*(8 choose 2)*(6 choose 2)*(4 choose 2) =
45*28*15*6 = 113,400.
The other way is 10!/2^5, which counts the number of ways of lining up ten people and divides by all the ways of switching the first with the second, the third with the fourth, and so on. This also gives 113,400.
But we know the intended answer is 945. So it must be that the order of the groups doesn't matter, i.e., any rearrangements of the five groups are not counted as different choices as long as the same pairs of people are still paired.
So we divide the other answer by 5!
113,400/120 = 945.