Question

How many electrons in an atom can have then $n=\text{ }5,l=2$ designation

Answer 1

Quantum number: The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers. There are four quantum numbers, namely, principal, azimuthal, magnetic and spin quantum numbers.

Complete step by step answer:
An orbital is the region of space around the nucleus within which the probability of finding an electron of given energy is maximum.
The probability at any point around the nucleus is calculated using the Schrodinger wave equation and is represented by the density of the points.
$n\text{ }=$ azimuthal quantum no telling about no of shell values of l can lie between $0$ or equal to zero and less than $n-1$ now, the d block contains a total $10$ columns of the Periodic Table.
Each group is equivalent to $1$ electron. This means that the d block, which is equivalent to the d subshell, can hold a total of $10$ electrons.
Therefore, a maximum of $10$ electrons can share the two quantum numbers $n=5,l=2$ value of m can be between equal to $-l$ and $+l$ when $n=5~$ $l=0,1,2,3,4$
and $m\text{ }=$for $l=2\text{ }m=-2,-1,0,1,2$ means $5d$ orbital and that can occupy upto $10$ electrons and so answer is $10$ electrons.
you can find the number of orbitals that can exist in a subshell by dividing the number of groups in a block by $2$ no. of orbitals in a subshell $=$ no. of groups in the block $2/2$
This is the case because an orbital can hold a maximum of $2$ electrons as stated by the Pauli Exclusion Principle.

Note: Orbital are the three dimensional structure in which the probability of finding an electron is maximum. In an orbit the number of orbits is equal to${{n}^{2}}$. And the number of electrons in it are equal to$2{{n}^{2}}$.
make sure you know well about Pauli Exclusion Principle