In the context of X-ray spectra, the emission lines observed are a result of electrons in an atom transitioning between different energy levels (or shells). Specifically, the lines known as K-lines arise when transitions occur between the K-shell (the innermost shell) and other shells.

The nomenclature for X-ray spectral lines includes the terms K_ɑ and K_β, which signify different types of transitions:

• K_ɑ line: An electron falls from the L-shell (n=2) to a vacancy in the K-shell (n=1).
• K_β line: An electron falls from the M-shell (n=3) to a vacancy in the K-shell (n=1).

Knowing this, let us evaluate the given options:

1. An electron transition from n=4 to n=2 does not involve the K-shell, so it is irrelevant to K-lines.
2. An electron transition from n=3 to n=2 also does not involve the K-shell, thus does not form a K-line.
3. An electron transition from n=3 to n=1 involves the K-shell and specifically results in a K_β line. Hence, this is the correct answer.
4. An electron transition from n=2 to n=1 involves the K-shell, but it results in a K_ɑ line, not a K_β line.

Thus, the correct answer is the transition of an n=3 electron to a vacant n=1 state, which generates the K_β line.