Step 1: Recall VSEPR theory.
According to Valence Shell Electron Pair Repulsion (VSEPR) theory, electron pairs surrounding the central atom repel one another and arrange themselves to minimize repulsion.
Step 2: Understand bond pairs and lone pairs.
Bond pairs are shared between two atoms and occupy less space around the central atom. Lone pairs are localized on a single atom and occupy more space because their electron density is concentrated closer to the nucleus.
Step 3: Compare bond pair - bond pair repulsion.
Since bonding electrons are shared between atoms, the repulsion between two bond pairs is the weakest among the three types of interactions.
Step 4: Compare lone pair - bond pair repulsion.
A lone pair occupies more space than a bond pair, so the repulsion between a lone pair and a bond pair is stronger than bond pair - bond pair repulsion.
Step 5: Compare lone pair - lone pair repulsion.
Two lone pairs repel each other most strongly because both have highly concentrated electron density around the central atom.
Step 6: Arrange the interactions in increasing order.
Therefore, the increasing order of repulsive interactions is
\[
\text{bond pair - bond pair}
<
\text{lone pair - bond pair}
<
\text{lone pair - lone pair}
\]
Step 7: Final conclusion.
Hence, the correct order is
\[
\boxed{\text{bond pair - bond pair} < \text{lone pair - bond pair} < \text{lone pair - lone pair}}
\]
Therefore, the correct answer is option (A).