No matter what color any player chooses, all multicolored permanents will be exiled.

For example, say you control a colorless creature, a blue creature, a green creature, and a creature that's both blue and green. No matter what you choose, your colorless creature will not be exiled, and your creature that's both blue and green will be exiled. If you choose blue or green, the creature you control that is only that color will not be exiled. It doesn't matter what anyone else chooses.

The colors chosen by players affect only whether or not permanents they control will be exiled. That choice does not affect whether particular permanents controlled by other players will be exiled.

Lands have no mana cost, so they are colorless unless an effect states otherwise. The colors of mana that a land can produce do not affect that land's color.

Players decide and announce in turn order, starting with the active player, which color they are choosing. Players later in the turn order will know which colors were chosen by players before them.