$$N$$ is an $$100000$$ digit number with no zero digit and the sum of the digits of $$N$$ is $$100001$$, then the number of such $$N$$s is

Each of the $$100000$$ digits is at least $$1$$, so their minimum total is $$100000$$. The required sum is only $$1$$ greater than this minimum, so exactly one digit must be $$2$$ and all the remaining digits must be $$1$$. The digit equal to $$2$$ can occupy any of the $$100000$$ positions, giving $$100000$$ numbers.