Does neutron decay have an additional factor of 2 in the decay rate?

In neutron decay, a down quark decays into an up quark. Does the fact that there are two down quarks with the possibility of decaying contribute a factor of 2 to the decay rate?

My confusion is that the two down quarks are in the initial state rather than the final state, whereas the density of states always considers the final state. However, my logic is that you can consider the neutron decaying into a proton where this down quark remains, and another where that down quark remains, and so the density of states is doubled.

Is this correct? I cannot find a discussion of it anywhere.