This is a familiar creationist argument, stated in this case by a non-creationist:

Consider the replacement processes needed in order to change each of the resident genes at L loci in a more primitive genome into those of a more favorable, or advanced, gene. Suppose that at each such gene locus, the argument runs, the proportion of gene types (alleles) at that gene locus that are more favored than the primitive type is K^−1. The probability that at all L loci a more favored gene type is obtained in one round of evolutionary “trials” is K^−L, a vanishingly small amount. When trials are carried out sequentially over time, an exponentially large number of trials (of order K^L) would be needed in order to carry out the complete transformation, and from this some have concluded that the evolution-by- mutation paradigm doesn’t work because of lack of time.

Basically, what creationists argue is that the evolution of new genes is linear and sequential — there is no history, no selection, it works entirely by random replacement of the whole shebang, hoping that in one dazzling bit of luck that the entire sequence clicks into the right sequence, and then it all works. If that were the way the process occurred, then they’d be right, and evolution would be absurdly improbable and would take an untenable length of time.

Another way to think of it is a bizarre version of the hangman guessing game, where one person thinks of a word, and the second person has to guess what it is. In the normal version of the game, the second person guesses letters one by one, and they’re placed in the appropriate spot. In the creationist version, you only get to guess a whole sequence of letters in each round, and you are only told if you are right or wrong, not which letters are in the correct position in the word. Not only does it become a really boring game, but it also becomes extremely unlikely that anyone can solve it in a reasonable amount of time.

Evolution does not work like that. It works in parallel, changing and testing each variant simultaneously in many individuals, and then selection for the most favorable subset of changes latches them in place, making the matching letters more likely to be fixed. Or, as the paper by Wilf and Ewens explains,

But a more appropriate model is the following. After guessing each of the letters, we are told which (if any) of the guessed letters are correct, and then those letters are retained. The second round of guessing is applied only for the incorrect letters that remain after this first round, and so forth. This procedure mimics the “in parallel” evolutionary process. The question concerns the statistics of the number of rounds needed to guess all of the letters of the word successfully.

That’s the question. If purely random changes would require a ridiculous length of time to match a target, proportional to K^L, how long would it take if we actually use more reasonable, biologically relevant model? Wilf and Ewens state the model in mathematical terms and derive a theoretical answer, and you won’t be surprised that it’s significantly shorter; you might be impressed at how much shorter the operation would take.

Instead of a time proportional to K^L, it will take a time proportional to K log L.

That’s very much shorter! To put some representative numbers on it, imagine a protein that is 300 amino acids long, made up of 20 possible amino acids, and I’m going to ask you to guess the sequence. Under the creationist model, you wouldn’t even want to play the game — it would take you on the order of 20³⁰⁰ trials to hit that one specific arrangement of amino acids. On the other hand, if you took a wild guess, writing down a random 300 amino acids, and I then told you which amino acids in which position were correct, you’d be able to progressively work out the exact sequence in only 20 log 300 trials, or around 50 guesses.

Notice that this is not a concrete estimate of the time it would take for something to evolve! It’s a grossly simplified version of the story: the example overstates the power of selection (amino acids won’t be locked in, but will only be less likely to change), and overstates the required accuracy of matching to a target (there would be more tolerance for variation), and the whole idea of meeting a specific target is not necessarily a good model. As a guide to short-circuiting the invalid assumptions of creationists, though, it’s handy to have a simple mathematical formula to remove that naive combinatorial model from the table.

Wilf HS, Ewens WJ (2010) There’s plenty of time for evolution. arXiv:1010.5178v1.