Can 3 Players Co-op on Xbox Live in Call of Duty: Black Ops (2010)?

No. The original Call of Duty: Black Ops (2010) supports:

• 2-player local splitscreen (offline + online with Gold).
• Online multiplayer up to 18 players (standard modes) – but no 3-player co-op campaign or zombies.
• Zombies: 1–4 players online, 1–2 splitscreen – but you need 4 separate accounts/consoles for 3+ players.

Workarounds

• Party up: 3 friends join same public match (different teams or same if slots allow).
• Custom Games: Host a private match with bots – up to 18 slots, but still 1 player per console.

Later Games With 3-Player Co-op

• Black Ops II – 4-player splitscreen zombies (2 per console × 2 consoles).
• Black Ops III / 4 / Cold War – 4-player online co-op campaign/zombies.
• Modern Warfare (2019) / MWII / MWIII – 3-player Spec Ops / Raids.

Stick to splitscreen duos or full lobbies on the 2010 title.