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The correct answer is $16$. The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let $x$ be the number of attendees. The expression $35+10.25x$ thus represents the total cost of the party. It's given that the budget is $\$200$, so this situation can be represented by the inequality $35+10.25x\le 200$. The greatest number of attendees can be found by solving this inequality for $x$. Subtracting $35$ from both sides of this inequality gives $10.25x\le 165$. Dividing both sides of this inequality by $10.25$ results in approximately $x\le 16.098$. Since the question is stated in terms of attendees, rounding $x$ down to the nearest whole number, $16$, gives the greatest number of attendees possible.