"With" is the same as "avec". Within means several things depending on context but all mean roughly "in" or "inside". When used to mean "inside" it is more formal or poetic than "inside". For example... "I was within the building" means the same as "I was inside the building" but is more formal. You might hear it in a court case or other formal setting. Another example: "You must pay this bill within three days." is more formal than "You must pay this bill in three days." but means the same thing. However one subtle confusing thing: "I am going to Paris in three days" means in *exactly* three days, but "I am going to Paris within three days" means "some time in the next three days". One common use that is not formal: "There are three Japanese restaurants within five miles." You also sometimes hear it used to mean the self, the person or the spirit, as in "Change comes from within." I hope this has not been too confusing!

The answers may lie within the document that the manager has with him.

Thank you Ruthi...I hadn't doubt for with but i see sometimes "within" in professionnal document (written by french people) and I didn't understand....with your explanation I am not sure that they use it correctly.

Within means the same as inside. It is a little old fashioned though and not really commonly used. With is a different preposition and like "avec" in French: I am with my friend right now. I will come with you. I drink my coffee with milk and sugar.