Consider the following statements in respect of the sum S = x + y + z, where x, y and z are distinct prime numbers each less than 10:

1. The unit digit of S can be 0.

2. The unit digit of S can be 9.

3. The unit digit of S can be 5.

Which of the statements given above are correct?

The correct option is 1 and 3 only.

Let us analyze the problem step-by-step to understand why statements 1 and 3 are correct, while statement 2 is incorrect.

The sum is given by:

$S=x+y+z$

where $x$, $y$, and $z$ are distinct prime numbers, and each is less than 10.

First, let us list all the prime numbers less than 10. These are:

2, 3, 5, and 7

Since $x$, $y$, and $z$ must be distinct, we need to choose any 3 prime numbers out of these 4 options and find their sum.

Let us write down all possible combinations of selecting 3 distinct prime numbers from {2, 3, 5, 7} and calculate their sums:

Combination 1: Choosing 2, 3, and 5

$S=2+3+5=10$

Here, the unit digit of $S$ is 0. This shows that Statement 1 ("The unit digit of S can be 0") is correct.

Combination 2: Choosing 2, 3, and 7

$S=2+3+7=12$

Here, the unit digit of $S$ is 2.

Combination 3: Choosing 2, 5, and 7

$S=2+5+7=14$

Here, the unit digit of $S$ is 4.

Combination 4: Choosing 3, 5, and 7

$S=3+5+7=15$

Here, the unit digit of $S$ is 5. This shows that Statement 3 ("The unit digit of S can be 5") is correct.

Looking at all four possible cases, the only possible unit digits for the sum $S$ are 0, 2, 4, and 5. The unit digit of $S$ cannot be 9. Therefore, Statement 2 is incorrect.

Thus, only statements 1 and 3 are correct.