Using Statement-I alone: Statement-I tells us that the sum of the three distinct prime numbers is less than 23. The possible prime numbers less than 23 are: 2, 3, 5, 7, 11, 13, 17, 19. The only triplet that satisfies this condition is (3, 5, 11), with the sum $3 + 5 + 11 = 19$, which is less than 23 and is prime. Hence, Statement-I alone is sufficient to answer the question.

Using Statement-II alone: Statement-II tells us that one of the numbers is 5.

• (5, 7, 11) has the sum $5 + 7 + 11 = 23$, which is prime.
• (5, 11, 13) has the sum $5 + 11 + 13 = 29$, which is prime.

With Statement-II, we have multiple possible sets of prime numbers (5, 7, 11) and (5, 11, 13), so Statement-II alone does not provide a unique answer.