I am fresh graduate and have received an offer letter from a company I recently got placed in. However their joining date coincides with my pre planned family vacation. I need an joining date extension of 10 days. Is it possible? What reasons should I give the HR?
To start with, of course its possible. Given that you are are college hire, I am guessing you haven't had a lot of control in choosing the appropriate date. Nevertheless, a heads up would have been nicer. But still, good companies don't penalize on such matters. Unless your position needs to be filled very urgently, there's good chance they will allow.
What reasons should I give the HR?
Nothing other than the truth. You don't want to start with the wrong foot. Earn their trust by being honest. 10 days IMO is not much. All the best.
In worst case, if they don't agree and you have to comply with them and postpone your plans, that would again be a good example of your reliability. Finally, take it all in good spirit.
You should have brought this up when they asked when you are able to start. Now you're in a position to look like you have some difficulty with planning, but if they just sent you an offer letter you still have time to save some face.
You should be honest and tell them that you have a long-standing planned vacation and, if truthful, tell them that it can't be changed and you won't be able to recover the money. Apologize and ask if it is possible to have your joining date be Y instead of X.
If you are being hired on individually it probably won't be a big deal, but if you're backfilling someone and there is limited time for knowledge transfer or if you are being hired in a group and are expected to do some kind of training/orientation, you may find yourself in a difficult position.
Whether it is possible or not will depend on the company's policies and plans. For example, some companies that hire a lot of new graduates do orientation and training for a block of new hires, making it more expensive to bring a new graduate on 10 days later.
The only way to know is to ask them. The obvious reason to give HR is the truth: "I have a pre-planned family vacation. May I delay my joining date to ....?".