I have the following question:
"If we have $n$ different kinds of objects such that we have an infinite amount of each kind, then in how many ways can we choose $k$ objects if only an odd number of each kind of object can be taken?"
I'd like to see if my reasoning checks out, as to me the part where "if only an odd number..." seems a bit tricky.
First, I must use generating functions, and choose to use the following result $$(1+x+x^2+\cdots)(1+x+x^2+\cdots)\cdots(1+x+x^2+\cdots) = \frac{1}{(1-x)^n}.$$
As the coefficient belonging to $x^k$ gives the number of ways of picking $k$ objects.
Furthermore, because we choose only odd numbers, then we can see that the number of ways of picking $k$ objects is given by the coefficient of $x^k$ of the following generating function $$(x+x^3+x^5+\cdots)(x+x^3+x^5+\cdots)\cdots(x+x^3+x^5+\cdots)=\frac{x}{(1-x^2)^n}.$$
Would summing over only the odd powers work?
Summing over the odd powers works well. Note that going from the first to the second generating function you have multiplied each term on the left by $x$, so the total left side by $x^n$. The numerator of the second should therefore be $x^n$