Let A, B, and C be the three mutually independent events with their probabilities equal to $x, y$ and $z$ respectively. Then the probability that exactly two of these events occur is

"Exactly two events occur" means we need to consider three separate cases:

1. A and B occur, but C does not
2. A and C occur, but B does not
3. B and C occur, but A does not

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Since the events are independent, we can multiply their probabilities directly.

Case 1: A and B occur, C does not

The probability that C does not occur = $1 - z$

Probability = $x \times y \times (1-z) = xy - xyz$

Case 2: A and C occur, B does not

The probability that B does not occur = $1 - y$

Probability = $x \times z \times (1-y) = xz - xyz$

Case 3: B and C occur, A does not

The probability that A does not occur = $1 - x$

Probability = $y \times z \times (1-x) = yz - xyz$

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The total probability that exactly two events occur:

$(xy - xyz) + (xz - xyz) + (yz - xyz)$

$= xy + xz + yz - 3xyz$

$= xy + yz + zx - 3xyz$

The answer is Option 3.