Evaluate the number of different numbers one can write, having at one's disposal n digits of a q-ary system.

Question

I am working on Zorich, Mathematical Analysis I and I am stuck on a problem in section 2.2. The problem asks me to evaluate the number of different numbers one can write with n digits of a q-ary system. The answer given is $q ^ {\frac{n}{q}}$, which surprises me because I was wondering if $q ^ n$ should be the obvious answer. Can somebody help? Thanks in advance!

Answer 1

That is part (b) of a three part question and I think interpretation of what the question is asking may need the whole question to be considered. Part (c) talks about $f(x)=x^{n/x}$ so the given answer to (b) is unlikely to an error but may be affected by what the words were intended to mean

Part (a) is

How many different numbers can one define using $20$ decimal digits (for example, two ranks with $10$ possible digits in each)? Answer the same question for the binary system. Which system does a comparison of the results favor in terms of efficiency?

and the answer to the first part is presumably $10^2=100$ as there are $10$ possibilities in the tens rank and $10$ possibilities in the units rank. With $20$ binary digits (ten ranks with $2$ possible digits in each) the answer is presumably $2^{10} = 1024$ making binary more able to represent different numbers

So in (b) the number of ranks is presumably $\frac{n}{q}$ (at least when $n$ is a multiple of $q$) and, with $q$ digits available for each rank, this makes the total number of possibilities $q^{n/q}$