Meaning of the coefficient of $x^r$ in this continued product

Question

Can someone please explain the 6th line of the explanation: "Also the coefficient of $x^r$ in the product is the number of ways of taking $r$ of the letters $a, b, c,\dots$"?

I cannot understand how the coefficient of $x^r$ in the continued product $(1+ax+a^2 x^2+\dots+a^p x^p)(1+bx+b^2 x^2+\dots+b^p x^p)(1+cx+c^2 x^2+\dots+c^p x^p)$ is the number of ways of taking $r$ of the letters $a, b, c,\dots$. How is it the number of ways? Also is there any relation between this concept and the distribution of balls into bins?

FYI, this seems to be an example of using a generating function. Since they can sometimes be quite useful in various fields, in particular such as combinatorics, you may be interested in reading the MSE post How can I learn about generating functions?. — John Omielan, Apr 07 '19 at 19:03

score 2 · Accepted Answer · answered Apr 07 '19 at 18:57

2

In order to calculate the $x^r$ coefficient we must select $r$ of the letters in a number of ways. This number of ways is given by the number of terms in the coefficient of $x^r$. For example, $$(1+ax+a^2 x^2+\dots+a^p x^p)(1+bx+b^2 x^2+\dots+b^p x^p)(1+cx+c^2 c^2+\dots+c^p x^p)$$ $$=1+(a+b+c)x+(a^2+ab+ac+b^2+bc+c^2)x^2+\dots$$ The number of terms in the $x^r$ coefficients are $1,3,6,\dots$ for $r=1,2,3,\dots$ and these values correspond to the number of ways to select $r$ of the letters $a,b,c$ with the restriction that the maximum number of $a$'s, $b$'s and $c$'s is $p$.

answered Apr 07 '19 at 18:57

Peter Foreman

19,947

So the author meant number of terms in the coefficient of $x^r$? – MrAP Apr 07 '19 at 19:01
I'm pretty sure, because $(a+b+c)$ is not equal to the number of ways to select $1$ of the letters $a,b,c$ - 3. – Peter Foreman Apr 07 '19 at 19:03

Meaning of the coefficient of $x^r$ in this continued product

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