Basically, 못 해요 means "I can't do". I assume the adult daughter meant something like "I can't do so, it's against my nature or principle" by that. You can say 안 돼 to mean "It doesn't work" when you can't do something. For example, 컴퓨터가 안 돼. The computer doesn't work. (I can't use the computer) -ㄹ 수 없습니다 is like "There's no way you(or we) can do (something)", here 수 means "capacity, way, method, ability". It's formal Korean. 잘 수 없습니다. I can't sleep. 고칠 수 없습니다. I can't fix it.

All three thing are not exactly the same. First of all 못 해요 and ~할 수 없습니다 are the same. '못 해요' should be used as an answer to the Y/N questions. '너 ~ 할 수 있어?' / '못해요'(No, i can't) on the other hand '~할 수 없습니다' can be used as declarative or questions '지금 컴퓨터를 사용할 수 있습니까?' '아니오, 지금 컴퓨터를 사용할 수 없습니다.' the base form is '~할 수 있다'(be able to ~) Second, 안 돼 means 'No, you can't' to the request. '엄마 지금 컴퓨터 써도 되요?' '안 돼!'

못하다 : no ability 할 수 없다 : no ability, not execute by situations like ability, knowledge, condition, mind. any reasons, etc... 안 돼 : 1. Don't do it. (maybe It is wrong for speaker.) 2. It can not be. BUT!!! you know... It is language.. so, speaker use all those. even if the usage is not proper. Usage 1) 할 수 없다 => 못한다 actually, he can do it. but He is telling so strongly. "나는 이것을 못해" grammatically, He should say, "나는 이것을 할 수 없어" or "나는 이것을 하기 싫어". I dont like to do it. because It doesn't feel like. Usage 2) 못한다 => 안된다 He doesn't have any ability to do it. But He said, "이건(이것은) 안된다." He should say, "이것을 못한다". Usage 3) 할 수 없다 => 안된다. He can not do it by many reasons. But, He might say, "이것은 안 된다." means "It doesn't be anything." or "It can not be anything". He should say, "이것을 할 수 없다". means "I can not do it" conclusion: those three are very similar when we tell about negative oneself's opinion for a thing. specially, 안돼 is two meaning. When we use those properly, It is depended on speaker's mind or opinion. So, After those conversation, the speaker may tell more details or if not, you can ask, :) 즐! but It is ok for speak if the usage make listener It is depenond on the

"못 해요"---means "you want to do sth but can not achieve the goal."(can not) "안돼"---means "you just dont want to do sth."(can not) "ㄹ 수 없습니다"----means"can not"

As Hailey mentioned, “수” in “ㄹ 수(가) 없다” means way/method/capacity, and sometimes it has replaced by other similar words like 도리/방법/길. 컴퓨터를 고칠 방법이 없다. / There’s no way to fix my computer. / I can’t fix my computer. 이 복잡한 쿼리를 더이상 최적화할 길이 없다. / There’s no way to optimize this complex query anymore. / This complex query cannot be optimized anymore.