When does your rental agreement begin? If it begins the day you move in, that's the day you start owing. If the agreement begins the day you signed or agreed to it orally, it begins that day, not the day you actually take occupancy. So understand which it is. If the agreement is that the room is yours - and is off the market to others - then you owe even if you won't be in for another three weeks.
UPDATE: Based on YOUR update, you have a rental agreement. It's your room. How much did you agree to pay for a full month? When is rent due? If it's due on the first of the month and you aren't moving in until the 22nd, or whenever, there is no reason to figure out a number for those unused days. You owe for all of them. You pay for the entire month. Then, the following month, when you will actually be living there, you pay for that month.