Find Coefficient of x^7 in (1+x+x^2+x^3...)^n: C(n+6,7)

[0rthodontist]

Problem: Find the coefficient of x^7 in (1 + x + x^2 + x^3 + ...)^n
I thought this should be C(n + 6, 6) since you have to distribute 7 x's between the n series with repetition. But my book says it is C(n + 6, 7). (C for combination) What is the explanation?

 

[matt grime]

Well, let's see what happens when n=0. The coefficient of x^7 is 0, and 6C6=1 (you answer) and 6C7=0 (their answer; it is a convention that it is zero).

You could prove it inductively, and it might (for once) give you an idea of what's going on (induction usually doesn't). Or just work it through for some small n to see what is happening.

 

[0rthodontist]

Ah, never mind, it was just a dumb error. I misused the formula for selection with repetition--mixed up my bins and my objects.