How do I randomly assign participants in a factorial experiment?

This is done essentially the same way as in an RCT, except instead of assigning people to 2 or 3 conditions, you need to assign people to many more conditions. One way to approach random assignment in a factorial experiment is to produce many sets of permutations of the integers from 1 to C where C is the number of conditions, and use these numbers to assign subjects to conditions in a permuted block fashion. For example, suppose you were conducting a 2³ factorial experiment, which of course has 8 experimental conditions, and you want to assign 160 subjects. You could use software to conduct random sampling of the integers 1 through 8 without replacement, and repeat this 20 times. You now have 20 lists of the numbers 1 through 8 randomly assorted, in other words, 160 random numbers. Then you can use these numbers to assign each of the 160 subjects. This approach will ensure that the experimental conditions fill up approximately evenly.

There are other approaches to random assignment in factorial optimization trials. We recommend the excellent article by Gallis et al. (2019).

References

Gallis, J. A., Bennett, G. G., Steinberg, D. M., Askew, S., & Turner, E. L. (2019). Randomization procedures for multicomponent behavioral intervention factorial trials in the multiphase optimization strategy framework: challenges and recommendations. Translational Behavioral Medicine, 9(6), 1047-1056.