Evaluating data sufficiency for determining an original price before discount

We require three distinct primes p_1, p_2, p_3 whose sum S = p_1 + p_2 + p_3 is also prime. For a sum of three integers to be an odd prime, the set must contain the only even prime, 2. If all three were odd, their sum would be odd + odd + odd = odd, which can be prime. Wait, let's re-verify the parity: Odd + Odd + Odd = Odd, so a set of three odd primes can sum to an odd prime (e.g., 3 + 5 + 11 = 19).

Let's test the statements: Statement I: S < 23. Possible combinations yielding prime sums < 23:

Is \{3, 5, 11\} the only one? What about \{2, 3, 5\} \rightarrow S = 10 (No). What about \{2, 7, 11\} \rightarrow S = 20 (No). What about \{2, 5, 7\} \rightarrow S = 14 (No). Thus, \{3, 5, 11\} appears to be unique under 23 if all elements are distinct primes. Let's double check if there are others. What about \{2, 3, 11\} = 16 (No). What about \{2, 5, 13\} = 20 (No).

Let's check if Statement I alone isolates \{3, 5, 11\}. If it does, Statement I is sufficient. Let's re-verify if any other triplet works. What about \{2, 7, 13\} = 22 (No). So \{3, 5, 11\} is the unique solution under 23. This would make Statement I alone sufficient, pointing to option (a).

Answer: a