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I was solving some problem related to generating functions ,and I have to find it's coefficient : So after some steps , I reached to this : $ ( 1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5) ( 1 + x + x^2) $
By Generating functions , $( ( 1 -x^5 /1 - x)$ * $ ( 1 -x^6/1 - x)$ * $ ( 1 -x^3/1 - x))$ ,
$( 1 -x^5 )( 1 -x^6)(1 -x^3 )/(1 - x)^3$ ,
Any help would be appreciated .

Johnathan
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  • Which term do you have to find the coefficient of? – B. Mehta May 20 '17 at 15:29
  • @B.Mehta , I had to find no. of ways to select 1 to 10 cars , from a pile of 4 Skoda , 5 Toyata and 2 Mercedes . – Johnathan May 20 '17 at 15:32
  • If you really have to find all the coefficients of $x^1$, $x^2$, ... $x^{10}$ then you might as well just multiply the original three polynomials. Just long-multiply $111111$ by $11111$ and then by $111$ BUT DON'T CARRY, do it in base $79$ or whatever. – ancient mathematician May 20 '17 at 16:21
  • Suggestion: Expand $(1-x^5)(1-x^6)(1-x^3)$ by simple multiplication, and write $(1-x)^{-3}$ as an infinite series by the binomial theorem. – awkward May 21 '17 at 00:18

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