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.