University asking for money for notice period

According to my offer letter it says "In the event either of us wishes to discontinue the association, it can be done after giving 3 months notice in writing to the appointing authority or by paying to the University 3 month's salary in lieu thereof. However, a faculty member will be required to serve with the University till the end of the academic semester during which he resignation is tendered. Relieving from the University duties will be processed on notice period end or semester end whichever date is later." I have tendered my resignation on 10th June which is in the academic session of Jan - June. And had mentioned in my resignation letter that I wish to get relieved in July end. Since, I have some leaves in my account, my 3 months notice period ending in 10th September would be taken care of. But the University is telling that they will not let me serve the notice period and I will be relieved on 30th June and I need to pay for the remaining days of my notice period. Sir, is this justified? I am not allowed to serve the notice period and I am being asked to pay the money. They are saying that if I serve the notice period then I would have to serve till the end of the next academic session which ends in December. But as per the above clause of my offer letter, I have tendered my resignation in this academic session Jan - June and I am suppose to serve only 3 months of notice period. Now they are not willing to allow me serve the notice period on the grounds that this is a lean period and there is no work. Why should I pay you. I am okay with that, no issues, release me in the month of June but why do I need to pay for the days that I am not being allowed to serve. Sir, is there any legal help in this matter or do I have to pay?