I have $20$ identical objects to distribute to $6$ different groups, but each group has to receive a different number of objects. How many ways to do it?
I know the way to distribute $n$ identical object to $r$ different groups is the same method as stars and bars formula $n+r−1 \choose n$.
But I don't know how to do with restriction "each group has to receive a different number of objects". (A group can receive no objects.)
Start off with $0+1+2+3+4+5=15$. The ways to add the remaining $5$ objects to this while retaining order and distinctness correspond to the partitions of $5$ (e.g. $5=2+3$ corresponds to $0+1+2+3+(4+2)+(5+3)$), of which there are $7$. You can assign each of the resulting partitions to the $6$ groups in $6!=720$ different ways, so the total count is $7!=5040$.