Total ways of selecting 6 people out of a group of 20 = 20C6 = 38760
But we have counted the case when we have either one couple, two couples or three couples. We will subtract these cases to get the desired result.
No of ways of selecting exactly one couple = 10C1 * 18 * 16 * 14 * 12 / 4! = 20160
(Reason, select a pair in 10C1 ways, now to select four more, there are 18 ways to select next person, after that we can't choose his/her partner, so for the third person we only have 16 ways, and so on 14 and 12, but here we have considered all permutations of 4 people, so to eliminate that we divide by 4!). Similarly for the other cases below,
No of ways of selecting exactly two couples = 10C2 * 16 * 14 / 2! = 5040
No of ways of selecting exactly three couples = 10C3 = 120
Answer = 38760 - 20160 - 5040 - 120 = 13440