Have a responsibility and have responsibility

Hello

Looking at the usage of "responsibility" I noticed that both "have responsibility" and "have a responsibility" are used. I wondered
1) if there is a difference in meaning between them, and

2) if there is a difference in usage as far as prepositions are concerned.

For example

Jenny, you have responsibility for clearing up the room after the class. (http://dictionary.cambridge.org/define.asp?key=67350&dict=CALD)

1) Is it equally fine to say: Jenny, you have a responsibility to clear up the room after class.

2) And how about the difference between the following sentences. Are both grammatically acceptable?

Jenny, you have a responsibility for clearning up the room after class.
Jenny, you have a responsibility to clear up the room after class.

Your help is appreciated very much.