If the distance is less than 10, then it must be something between 0 and 9.
You can create all possible 100 bit patterns from one bit pattern A by choosing the bits from A that change.
That means that you need to choose less than 10 bits that change. More precisely, you need to choose exactly n bits that change, where .
The number of ways to do this is
The total number of 100 bit patterns is .
Hence, the required probability is
assuming that each bit pattern is chosen with equal probability.