Consider the following sets of quantum numbers.
(i) `{:(n,l,m,s,),(3,0,0,+1//2,):}`
(ii) `{:(n,l,m,s,),(2,2,1,+1//2,):}`
(iii) `{:(n,l,m,s,),(4,3,-2,-1//2,):}`
(iv) `{:(n,l,m,s,),(1,0,-1,-1//2,):}`
(v) `{:(n,l,m,s,),(3,2,3,+1//2,):}`
Which of the following sets of quantum number is not possible ?

Options

A. (ii), (iii) and (iv)
B. (i), (ii), (iii) and (iv)
C. (ii), (iv) and (v)
D. (i) and (iii)

Explanation

To determine which sets of quantum numbers are not possible, we need to apply the rules governing quantum numbers:

1. **Principal Quantum Number (n)**: This can take any positive integer value (1, 2, 3, ...).
2. **Azimuthal Quantum Number (l)**: This can take integer values from 0 to (n-1).
3. **Magnetic Quantum Number (m)**: This can take integer values from -l to +l, including 0.
4. **Spin Quantum Number (s)**: This can take values of +1/2 or -1/2.

Now, let's analyze each set of quantum numbers provided in the question:

### Set (i): (n, l, m, s) = (3, 0, 0, +1/2)
- **n = 3**: Valid, as n is a positive integer.
- **l = 0**: Valid, since l can be from 0 to (3-1).
- **m = 0**: Valid, since m can be from -0 to +0 (only 0).
- **s = +1/2**: Valid, as s can be +1/2 or -1/2.

**Conclusion**: This set is valid.

### Set (ii): (n, l, m, s) = (2, 2, 1, +1/2)
- **n = 2**: Valid.
- **l = 2**: Invalid, since l must be in the range 0 to (2-1) = 1.
- **m = 1**: Not checked, as l is already invalid.
- **s = +1/2**: Not checked, as l is already invalid.

**Conclusion**: This set is invalid.

### Set (iii): (n, l, m, s) = (4, 3, -2, -1/2)
- **n = 4**: Valid.
- **l = 3**: Valid, since l can be from 0 to (4-1).
- **m = -2**: Valid, since m can range from -3 to +3 (l = 3).
- **s = -1/2**: Valid.

**Conclusion**: This set is valid.

### Set (iv): (n, l, m, s) = (1, 0, -1, -1/2)
- **n = 1**: Valid.
- **l = 0**: Valid.
- **m = -1**: Invalid, since m must be in the range -0 to +0 (only 0).
- **s = -1/2**: Not checked, as m is already invalid.

**Conclusion**: This set is invalid.

### Set (v): (n, l, m, s) = (3, 2, 3, +1/2)
- **n = 3**: Valid.
- **l = 2**: Valid.
- **m = 3**: Invalid, since m must be in the range -2 to +2 (l = 2).
- **s = +1/2**: Not checked, as m is already invalid.

**Conclusion**: This set is invalid.

### Summary of Validity:
- Set (i): Valid
- Set (ii): Invalid
- Set (iii): Valid
- Set (iv): Invalid
- Set (v): Invalid

### Final Answer:
The sets of quantum numbers that are not possible are (ii), (iv), and (v).