A gaming device having an accumulation game with changing selections. The gaming device generates or picks, one at a time, a plurality of selections. Each generated selection is accumulated or marked and then changed to another selection. The gaming device continues generating selections, accumulating the generated selections and changing the generated selections until at least one selection is generated and accumulated a designated number of times. When at least one selection is generated and accumulated the designated number of times, an award is provided to the player wherein the award is based on the selection which was generated and accumulated the designated number of times.