One thing to recognize immediately, is that in the course of making permutations or combinations we are really just making different kinds of subsets . Here we emphasize that choosing and making subsets are really the same operation. Here we have a use for both combinations and permutations, since we first need to choose 2 places from 7 to hold the vowels, and then figure out the number of ways to arrange the vowels and the five consonants. Each term in the expansion shows a different way to distribute the 3 objects into two subsets, with the exponent of a showing the number of objects in one subset, and the exponent of b showing the number of objects in the second subset. The sum of the exponents always equals n, of course.
Every term is the sum of the terms flanking it in the row above.
Each row is a palindrome. Second number in each row tells you the row number. Every term except the first and the last is a multiple of n iff n is prime. Distribution of odd and even numbers seems predictable. We can prove this by induction, of course.