Question

Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for...

Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom? -4, 3, 1, 1/2 4, 2, 3, -1/2 2, 1, -2, 1/2 2, 1, 0, -1 3, 4, 0, 1/2 5, 3, 0, 1/2 5, 3, -3, 1/2 2, 1, 0, 1/2

The answer to your question is determined by using the definitions of the 4 quantum numbers.
n = 1, 2, 3, ..., n
l = 0 to n − 1. This includes all possible values up to n-1.
ml = all values from -l to +l. Example if l = 2 them ml = -2, -1, 0, +1, +2
ms = +1/2 and -1/2

-4, 3, 1, 1/2 - not possible because n cannot be negative.

4, 2, 3, -1/2 - not possible because ml must be from -2 to +2 by definition.

2, 1, -2, 1/2 - not possible because ml must be from -1 to +1 by definition.

2, 1, 0, -1 - not possible because ms can equal only +1/2 and -1/2.

3, 4, 0, 1/2 - not possible beacuse l cannot be greater than n.

5, 3, 0, 1/2 - This is possible according to all of the definitions.

5, 3, -3, 1/2 - This is possible according to all of the definitions.

2, 1, 0, 1/2 - This is possible according to all of the definitions.