**The Quantum Numbers**
The quantum numbers describe the characteristics and properties of an electron in an atom. There are four quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms). Each quantum number has a specific range of values.

**Principal Quantum Number (n)**
The principal quantum number (n) determines the energy level or shell in which the electron is located. It can have positive integer values starting from 1, representing the different energy levels in an atom.

**Azimuthal Quantum Number (l)**
The azimuthal quantum number (l) defines the shape of the orbital. It can have values ranging from 0 to (n-1) for a given value of n. The values of l determine the subshells within a particular energy level.

**Magnetic Quantum Number (ml)**
The magnetic quantum number (ml) specifies the orientation of the orbital in space. It can have integer values ranging from -l to +l, including zero.

**Spin Quantum Number (ms)**
The spin quantum number (ms) describes the spin orientation of the electron. It can have two possible values: +1/2 (spin-up) or -1/2 (spin-down).

**Analyzing the Given Options**
Let's analyze the given options one by one and determine if they represent the correct set of quantum numbers for the last electron of a Pd (Palladium) atom:

1) 5,0,0,1/2: This option represents a principal quantum number (n) of 5, which indicates that the electron is in the fifth energy level. The azimuthal quantum number (l) is 0, indicating an s orbital. The magnetic quantum number (ml) is 0, which means the orbital is not oriented in space. The spin quantum number (ms) is 1/2, representing a spin-up electron. This set of quantum numbers is not correct for the last electron of a Pd atom.

2) 4,2,0,-1/2: This option represents a principal quantum number (n) of 4, indicating that the electron is in the fourth energy level. The azimuthal quantum number (l) is 2, indicating a d orbital. The magnetic quantum number (ml) is 0, which means the orbital is not oriented in space. The spin quantum number (ms) is -1/2, representing a spin-down electron. This set of quantum numbers is not correct for the last electron of a Pd atom.

3) 4,2,3,1/2: This option represents a principal quantum number (n) of 4, indicating that the electron is in the fourth energy level. The azimuthal quantum number (l) is 2, indicating a d orbital. The magnetic quantum number (ml) is 3, which is not a valid value for a d orbital. The correct range of ml values for a d orbital is -2, -1, 0, 1, or 2. This set of quantum numbers is not correct for the last electron of a Pd atom.

4) 4,2,3,-1/2: This option represents a principal quantum number (n) of 4, indicating that the electron is in the fourth energy level. The azimuthal quantum number (l)