All depends on how you define "random". A sequence of M 1's followed by N-M 0's is as random as any other sequence, as they will have the exact same probability of being generated by a truly random random-number generator.

I remember Knuth goes quite detailed into this random-sequence thing in Volume 2.

My feeling is that it is more about "predictability": once a random sequence is generated, the randomness is gone and all you have is a sequence. The thing is that you should have no way to predict what sequence will be generated. If you put additional constraints (such as x 1's) then as soon as you have seen x 1's being generated, you know for sure that from then on only 0's will follow.