A gaming device that enables players to accumulate awards by activating an award distributor having a plurality of award symbols and at least one selection group activator symbol. The gaming device provides the player with a plurality of activations where an award is associated with each award symbol indicated in each activation. When a selection group activator symbol is indicated, the gaming device displays at least one selection set having a plurality of selections associated with selection awards. The gaming device enables a player to select one selection and provides the associated selection award to the player. The number of available selections in the selection set decreases by one after the player picks a selection. If the selection group activator symbol is subsequently indicated, the player picks from the remaining available selections in the selection set.