1. 

Suppse we have two combinatorial families with no overlap between the two, where the size of A is a(n) and the size of B is b(n). We create a new set C=AXB. To define the size of C or c(n) in terms of a summation defined by variables n and k. n is a(n)'s upper limit of A's possible size value. k is the indexing value of the summation ranging from 0 to n. If k is the current size of A, the the size of B is n-k. Thus, a(k) is the number of ways to get k objects from A and then b(n-k) is the number of ways to get n-k objects from B. Since we have n possibilities to choose from, there are n choose k or binomial(n,k) ways to pick k from A. Thus, c(n) = Sum(binomial(n,k)*a(k)*b(n-k), k = 0..n). We multiply both sides by x^n/n! to convert c(n) to its egf. Simplifying, we see that egf(C)=egf(A)*egf(B). Iterating, we get egf(A1xA1x...xAk)=egf(A1)...egf(Ak).


2. 

ordinary generating function
sum(n*(n-1)*(n-2)*x^n,n=0..infinity) =
6*x^3/(x - 1)^4

exponential generating function
sum(n*(n - 1)*(n - 2)*x^n/n!, n = 0 .. infinity) = 
x^3*exp(x)


3.

(i)
coeff(taylor(egf,x=0,101), x, 100) * 100! = 
86635961604145793709294621186421021577187015751602637085427283871117865337554218116225569701173834817160231371909161518471870

(ii)
coeff(taylor(exp(exp(x) - 1)*exp(exp(x) - 1), x = 0, 101), x, 100)*100!/2! =  43317980802072896854647310593210510788593507875801318542713641935558932668777109058112784850586917408580115685954580759235935