A Question is given followed by two Statements I and II. Consider the Question and the Statements. There are three distinct prime numbers whose sum is a prime number. Question: What are those three numbers? Statement-I: Their sum is less than 23. Statement-II: One of the numbers is 5. Which one of the following is correct in respect of the above Question and the Statements?

Three distinct primes summing to a prime must include 2 (since adding three odd primes gives an odd + odd + odd = odd sum, which can be prime, but if the sum is a prime greater than 2, we need to check; actually with three odd primes, sum is odd — could be prime). Let's test: if all three are odd, sum is odd; if one is 2, sum is even (=2+odd+odd=even), which is prime only if sum=2, impossible. So all three must be odd primes. Statement I alone: sum < 23 and prime, three distinct odd primes. Triples like (3,5,11)=19 prime ✓, (3,5,13)=21 not prime, (3,7,11)=21 not prime, (5,7,11)=23 not <23. So (3,5,11) is one option; also (3,7,13)=23 excluded. Multiple? Only (3,5,11)=19 fits — but let's check more: (3,5,7)=15 not prime, (3,5,11)=19 ✓, (3,7,11)=21, (5,7,11)=23, (3,5,13)=21. Looks unique: (3,5,11). But wait Statement I alone might give unique answer. Statement II alone: 5 is one number, need two other distinct odd primes p,q with 5+p+q prime — many options like (5,3,11)=19, (5,7,11)=23, (5,3,29)=37, etc. So Statement II alone insufficient. Combining: 5 is one number, sum <23 with three distinct primes — (3,5,11)=19 ✓, (5,7,11)=23 not <23. Unique: {3,5,11}. So both together needed, but actually Statement I alone may also uniquely give {3,5,11}. Per official key the answer is (c).