There are n balls of different colors and N lattices (n = N) of different sizes. The lattices are labeled as 1, 2, ..., N. Suppose that each ball falls into a lattice with equal probability and each lattice can hold arbitrarily many balls. Now, n balls are put into N lattices at random.
Find the total number of ways to have the first n lattices containing one ball each (i.e., the lattices with label 1,2,...,n) and the other N-n lattices containing no balls.
Find the total number of ways to have n lattices containing one ball each, and the other N-n lattices containing no balls. Note that in this part, the n lattices may be any n lattices out of the N lattices in total.
Please explain your processes.