The particles that move around the nucleus in constant and random motion are electrons.

Here's why:

* Electrons are negatively charged: They are attracted to the positively charged nucleus.

* Quantum mechanics: Electrons don't orbit the nucleus in a classical sense like planets around a star. Instead, they exist in a cloud of probability called an electron cloud, where their exact location is uncertain.

* Constant motion: Electrons are constantly in motion within this electron cloud.

* Random motion: The movement of electrons is not predictable or deterministic. We can only describe the probability of finding an electron at a certain point in space.

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