っこない means "there's no way": the speaker is sure something can't or won't happen. It goes on a verb's ます-stem, and very often on the stem of a potential verb – , , .
、。
There's no way I can solve such a difficult problem.
。
There's no chance that person will keep their promise.
。
Even if you run now, there's no way you'll make it in time.
The verb is in the stem, not the dictionary form: ×できるっこない, ×勝つっこない. っこない almost always stays as it is at the end of the sentence; a past form like exists but is rare.
っこない, わけがない and はずがない
and
also mean "there's no way", and they work in any register. っこない is casual only, and it sounds more personal – exasperated, dismissive, or reassuring among friends. In polite speech or writing, use one of the other two.
。
There's no way that can be done.