The provided answer to the question, option 1, states that the order of increasing penetrating power in a given medium is a-particles, P-particles, and y-rays. Let`s dissect each option to understand why option 1 is correct.

Option 1: a, [3, y

This option suggests that a-particles have the lowest penetrating power, followed by P-particles, and then y-rays. This is consistent with the properties of these particles. A-particles are large and heavy, and they interact strongly with matter, causing them to have lower penetrating power. P-particles, also known as protons, are smaller and lighter than a-particles, but still have enough mass to interact with matter to a significant extent. Y-rays, on the other hand, are electromagnetic radiation and have no mass, allowing them to pass through matter with the least interaction.

Option 2: (3, y, a

This option suggests the opposite order of increasing penetrating power, stating that y-rays have the lowest penetrating power, followed by P-particles, and then a-particles. This contradicts the properties of these particles, as we know that y-rays have the highest penetrating power among the three.

Option 3: a, y, P