Design is like painting. A painter has a palette of different colors to choose from, and a final painting is typically a combination of different paints from the palette.
Of course, a painter could use just one color. But if the canvas was large and the products diverse, it's pretty unlikely that a painter would use one color exclusively.
It's even more unlikely that the painter would use one specific color, like burnt sienna. However, if we knew of a painter who had no choice but to use burnt sienna, then a painting in burnt sienna would be far more likely to be by this restricted artist than by any other painter.
The above is an analogy to the argument I have given for why evolution disproves design. Neo-Darwinian evolution places a lot of constraints on the way things have to be. There has to be descent (breeding) and common descent (any two individuals have a common ancestor). Experiment tells us that common descent is true. In other words, something like neo-Darwinian evolution has been used consistently throughout the history of life on this planet.
Evolution is just one method of design. In general, a designer doesn't need to try every possible design. A designer can simulate and use complex rules to create a single design, then just implement that design by manufacturing, with no need for breeding. In general, a designer can also create designs that have little in common with past designs, and which are constructed from completely new materials.
Since the number of ways of designing things without evolution is far greater than the number of ways of designing things with evolution, the fact we see evolution and only evolution means that a designer is ruled out.
This is my standard probability argument against design.
One response I get back from intelligent design sympathizers is that evolution is a design technique that even we humans use. Clearly, anyone who responds this way is missing the point entirely. I say from the very beginning that evolution is a form of design. The point is that it is just one form of design, and there's no reason to use just one technique exclusively. If there was a reason for exclusive use of evolution, and if there were any rational reason to grant that scenario a privileged share of the probability, then there would be predictions stemming from that reason.
However, it occurred to me that there's yet another good reason to dismiss the ID rebuttal. Suppose I'm designing an airliner. I don't want my airliners to breed for survival. Airliners are supposed to be useful to me, not to themselves. However, suppose I want to use evolutionary methods to design the wing of the airliner. That's aplausible scenario. So I create simulations of wings, test them in virtual environments, and preferentially carry forward effective designs in the simulation, and cull ineffective designs from the population. This is a straightforward application of genetic algorithms to the problem of wing design.
However, at no time in my simulation to I create a detailed mechanism for the wings to breed or have sex. There is, of course, a mechanism for simulating the outcome of "breeding" wings, i.e., of passing on design elements or altering population sizes. However, none of the wings have genitals, and they don't shag.
So, even if a designer chose to develop life on this planet using evolutionary techniques, there's no need for breeding. Thus, the fact that we have genitals is yet another reason why we're not designed. If we were designed (or if we're the evolutionary simulation), we wouldn't need genitals.