Randomly Organized Hair Particles

Hey,
Does anyone know if there is an easy way to randomize which objects are emitted while keeping an even distribution?
I am playing with some color ideas for a 3D printed Catan board and set my particle system to have one object per face, jittered and with even distribution. But if I select randomize it overrides the “one object per face” option.
Under the render setting you can also have it pick random but then it does not always have the right number of each object. I even tried using the “Use Count” option but it did not seem to help with that.
Having a collection with the right number of each tile and using that to determine how many of each one there was sort of works but then I cannot play with the seed value. I feel like there is some small thing that would allow me to randomize while keeping one per tile location.
Any thoughts?

Thanks in advance!!
Catan Colors|690x388