Let's just use different technique to calculate this problem. The total number of ways that 2 people can be selected from all the people available irrespective of the groups is 24C2. 24 because 8+6+10=24.
This is the total number of ways to select two people out of 24 people.
We have to select two people from all these groups such that they are from different groups. Instead of calculating this we can calculate that both the people chosen are form the same group and then subtract this from the total.
So to select two people from the same group we get:
8C2 + 6C2 + 10C2 here there is a plus sign because we have the or condition.
Either the two people are from the first group or the second or the third.
So our answer should be 24C2 - (8C2 + 6C2 + 10C2) = 188.