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It does- I remember enough from Linear Algebra that vector spaces aren't completely foreign, which helps a lot.

I think the generating-function method[1] is a case of pulling out a powertool (generating functions) to prove a simple thing to demonstrate the power of the tool, whereas your explanation takes the direct route and connects more of the threads about why it's true.

[1] found an instance here: http://austinrochford.com/posts/2013-11-01-generating-functi...

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