Definition of coordination number

The coordination number of the central metal atom/ion in a complex is the number of sigma bonds formed by ligands with the central metal atom.

In other words, it is the number of donor atoms directly attached to the metal through coordinate bonds.

Evaluate each option

Option A: the number of ligands around a metal ion bonded by sigma bonds

• This matches the definition of coordination number.
• Hence, A is correct.

Option B: the number of only anionic ligands bonded to the metal ion

• Coordination number does not depend only on anionic ligands. Neutral ligands like $NH_3$, $H_2O$, and CO also contribute.
• Hence, B is incorrect.

Option C: the number of ligands around a metal ion bonded by sigma and pi-bonds both

• Coordination number is counted based on direct sigma bonding donor sites, not on additional $\pi$-bonding interactions.
• Hence, C is incorrect.

Option D: the number of ligands around a metal ion bonded by pi-bonds

• Coordination number is not determined by only $\pi$-bonds.
• Hence, D is incorrect.

Final answer

The coordination number is determined by the number of ligands (more precisely, donor atoms) directly bonded to the metal through sigma bonds.

Therefore, the correct option is:

$\boxed{A}$

Comparison with stored correct answer

Stored correct answer = A
Derived answer = A
So, the derived answer agrees with the stored answer.