I am studying now the concept of generating function, and I have a solved question in my book which I don't understand, completely. There it is:
What is the number of options to roll 10 different dice, so the sum of the results will be 25?
Now for the solution:
f(x) = (x+x^2+x^3+x^4+x^5+x^6)^10
I believe this is because we have 6 options in each dice, and 10 different dice.
Here the professor took out the Common factor x^10 (I think by taking out x^6 and some more)
=x^10(1-x^6)^10 * (1+x+x^2....)^10
The "...." sign is a sign for an infinite arithmetic progression or until x^5? I don't know.
Now we have to find the "base" of x^25, which is the basic of a generatic function or so I believe. Now he makes so permutations here:
C(10+15-1,15) - 10*C(10+9-1,9) + C(10,2)*C(10+3-1,3) = =C(24,9)-10*C(18,9)+36*C(6,3)= =1,307,504 - 486,200+720 = 822,040
I believe it has to be related to this concept, that the book shows for Derivative. For example, f'(x)=n/(1-x)^(n+1); f''(x)=n(n+1)/(1-x)^n+2 Now he shows for the k-th Derivative that: 1/k! * f'(x) in the k-th time, if we set x=0:
n(n+1)....(n+k-1) / k! = C(n+k-1,k)
I am sorry for the long, not so understandable question, but It's really hard for me to understand his methods.