Three type

I wrote the following phrase in my PhD thesis:

"the following three general types of component" 

It would also be acceptable to write 'components', in the plural. I don't know if there's a prescriptive rule on the matter but speakers seem to feel that both are OK. So I ended up with having 'types' in the plural and the type of thing I was talking about, 'components', in the singular, and that's the way it stands. I think it sounds more academic or formal that way, whereas perhaps I'd use the plural in speech. I don't know; I'd have to check. 

But I was tempted, just for a moment, to write this:

'the following three general type of component'

Here, 'type' is singular, and so is 'component'. In combination with 'three', indicating that I'm not talking about just one type of component, this is definitely not grammatical for me. Nevertheless, my little addled brain wanted to write it and really thought it was all right for the moment. I don't have a reason why, and maybe there isn't one because after a little consideration I dismissed it. Certainly, having 'general' intervening between the number and 'type' made it easier to do, and leaving it out makes the singular 'type' sound much worse.

But it is also true that singular/plural agreement with 'type' (and related words like 'kind') is hard.

(If you're interested, this Stack Exchange thread discusses it, and doesn't come to any conclusion other than that maybe the one with agreement where both are either singular or both are plural is more natural.)