You must pay the fees ______ a week.

Concept: Prepositions of time link an action to a time reference. When a sentence sets a time limit by which something must be completed, “within” + a duration means “at any point inside that span of time” — the action may happen anywhere up to and including the end of the period.

Application: “You must pay the fees ______ a week” states a deadline: the payment has to be made at some moment during the next seven days. “within a week” fits exactly — it means “before the seven-day span runs out,” i.e. any time inside that one-week window.

Contrast — why the other prepositions fail:

• “after” + a week → places the action once the week has already passed, which reverses the deadline (it would mean you pay only when a week is over, not during it).

• “before” + a week → “before” needs a fixed point in time (“before Friday”); a bare duration like “a week” cannot follow it naturally, so the phrase is not idiomatic for a deadline.

• “by” + a week → “by” marks a specific cut-off point (“by Monday,” “by next week”). “by a week” has no fixed point of reference, so it is ungrammatical here.

Result: “within” is the only preposition that correctly expresses “at any time inside a one-week period.”