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?

The correct option is: The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

Analysis of the Problem:
We are given that there are three distinct prime numbers, let's call them $p$, $q$, and $r$, such that:
$p<q<r$ (without loss of generality, to keep them distinct)
And their sum is also a prime number, say $S$:
$S=p+q+r$, where $S$ is a prime.

Important Property of Primes:
All prime numbers except 2 are odd.
If all three distinct prime numbers $p$, $q$, and $r$ were odd, then their sum $S$ would be the sum of three odd numbers:
$\text{Odd}+\text{Odd}+\text{Odd}=\text{Odd}$.
This is possible since the sum $S$ (which is prime) must also be odd (since the smallest possible sum of three distinct odd primes is $3+5+7=15$, and any prime sum must be greater than 2).
However, if one of the three prime numbers is even (which must be 2, the only even prime), then the sum $S$ of two odd primes and one even prime (2) would be:
$\text{Even}+\text{Odd}+\text{Odd}=\text{Even}$.
Since the sum $S$ must be a prime number greater than 2, it cannot be even.
Therefore, 2 cannot be one of the three prime numbers.
Thus, all three prime numbers $p$, $q$, and $r$ must be odd primes.

Analyzing Statement-I: Their sum is less than 23.
We need to find three distinct odd primes $p<q<r$ such that their sum $S=p+q+r$ is a prime and $S<23$.
Let us test the smallest distinct odd primes: 3, 5, 7, 11, 13, 17, 19.
The minimum possible sum of three distinct odd primes is:
$3+5+7=15$ (15 is not prime).
The next possible combinations are:
1) $3+5+11=19$ (19 is a prime number).
2) $3+7+11=21$ (21 is not prime).
Any other combination of three distinct odd primes will result in a sum greater than or equal to:
$3+5+13=21$ (not prime), or $3+7+13=23$ (which is not less than 23).
Thus, the only unique set of three distinct prime numbers whose sum is a prime number less than 23 is 3, 5, and 11.
Since we get a unique set of three numbers, Statement-I alone is sufficient to answer the question.

Analyzing Statement-II: One of the numbers is 5.
If one of the numbers is 5, we can have multiple sets of three distinct primes that sum up to a prime. For example:
- Set 1: {3, 5, 11} where the sum is $3+5+11=19$ (19 is prime).
- Set 2: {5, 7, 19} where the sum is $5+7+19=31$ (31 is prime).
Since there are multiple possibilities for the three numbers, we cannot determine a unique set of three numbers.
Thus, Statement-II alone is not sufficient.

Conclusion:
The question can be answered by using Statement-I alone, but cannot be answered using Statement-II alone.