Informally I agree with you. But more formally it's not clear that no pattern exists in 3478907834617856 simply because one is not obvious.

I like the standard test of significance approach better. Assume that the numbers are generated by a discrete uniform distribution and then calculate the chance of a sample that extreme. More exactly we could let f(sample) = #_of_different_digits and see that f(sample) = 1 is a rare, extreme value: only 1/10^9 samples give f(sample) = 1. That makes our assumption (intuitively) less likely, though admittedly still possible.