A previous question pertains to a formula for the total number of points in a 3d discrete coordinate system that each total number of coordinate digit lengths $l$ can describe.

Is it possible to construct such a formula for a general $n$-dimensional case?

The coordinate digit length is defined as follows

$l=\lfloor log(x_1) \rfloor + ... + \lfloor log(x_n) \rfloor+n$.

where $x_1,...,x_n\in$ natural numbers.

In other words, I am looking for a formula that outputs the total number of positive integer points in a $n$-dimensional space that a certain total coordinate digit length can describe. For example, how many points can a total coordinate digit length of $5$ describe in a $10$-dimensional space?


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