There are $n$ distinct sets, and each sets has many similar objects. I have to choose $r$ objects from these sets. Any number (including all $r$) of objects can be picked up from any set. My solution goes like this:
Take $r$ objects from each of the $n$ sets and put them together. So now we have a set with $nr$ objects with $r$ objects of each kind.
Choose $r$ objects from this set. This can be done in $\dfrac{{}^{nr}C_r}{(r!)^n}$ ways. The denominator is there to make for the equivalence of $n$ sets of $r$ objects.
This derivation is wrong as can be verified with a small example. According to a book the correct answer is ${}^{n+r-1}C_r$. This formula gives the correct answer.
What is wrong with this derivation and how can the correct answer be derived?
What is wrong is your symmetry factor. You should only divide by the factorial of the objects used which is a varying number.
One way to obtain the correct answer is to place the $r$ objects in a line and then distribute $n-1$ separators between them. Objects between two separators will then come from the same set. It remains to count the ways to mix $r$ objects and $n-1$ separators which is the binomial coefficient.