Show that the number of partitions of an integer n into parts the largest of which is r is equal to the number of partitions of n into exactly r parts.
Solution. We are trying to find a way to relate two different types of partitions of n both in terms of r. Perhaps a Ferrers diagram could lead us in the right direction. So let us try some examples.
Suppose n = 10 and r = 3. Then one partition of n in which r is the largest part is 3+3+2+1+1. In a Ferrers diagram this looks like: