Selection of objects with generating functions

Question

Use generating functions to find the number of ways to choose $r$ objects of $n$ different types, knowing that we must choose at least 1 object of each type.

How can we express in the solution that the objects are different?

Answer 1

Assuming the question is:

There are $n$ different types of objects, how many ways can we choose $r$ objects, at least one of each type?

Then, you want the coefficient of $x^r$ in $$\left(x+x^2+x^3+\cdots\right)^n = x^n\left(1+x+x^2+\cdots\right)^n=\frac{x^n}{(1-x)^n}$$