How many orbitals are found in the d sublevel?

We can derive this from knowing that the atomic orbitals each exist in accordance with angular momentum quantum number `l`.

Recall how `l` tells you the shape of the atomic orbital. That just means:

`l = 0 -> s` orbital
`l = 1 -> p` orbital
`l = 2 -> d` orbital
etc.

Additionally, what we have associated with `l` is the magnetic quantum number `m_l`, the projection of `l` in the negative, unsigned, and positive directions. In other words...

`m_l = 0, pm 1, pm 2, . . . pm (l-1), pm l`

If `l = 2`, then:

`color(blue)(m_l = -2, -1, 0, +1, +2)`

Each individual `m_l` value corresponds to a unique orbital.

That indicates that five orbitals are available in the `d` orbitals. Specifically, the `d_(z^2)`, `d_(x^2 - y^2)`, `d_(xy)`, `d_(xz)`, and `d_(yz)` orbitals.

Furthermore, from knowing that the spin quantum number `m_s` for an electron is `pm"1/2"` (two spin states), and recalling the Pauli exclusion principle (two electrons in one orbital must be opposite spins), there can be a max of two electrons per orbital.

Therefore, the total number of electrons in five orbitals can be a maximum of 10.