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So I'm trying to work through some combinatorics problems and am struggling really hard since the book doesn't explain very well or give very concrete examples..

The question is asking "for each of the following expressions, list the set of all formal products in which the exponents sum to 4"

a) $(1+x+x^3)^2 (1+x)^2$

I've looked at the back of the book and I have no idea how they came up with their answer.. or why $x^3$ isn't used when trying to sum to $x^4.$

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    Expand the terms to get $(1+x^2 +x^6 + 2x + 2x^3 + 2x^4)(1+2x+x^2)$. – 2017-02-16

1 Answers 1

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The different ways of summing to 4 are 0+4, 1+3, 1+1+2, 1+1+1+1, 2+2.

You can write your product as $ (1+x+x^3)(1+x+x^3) (1+x)(1+x)$.

How many ways can those sums occur in this product?

For example, $x^3x = x^4$ occurs 4 times: each of the $x^3$ with each of the $x$.

Another example: 1+1+1+1 occurs only once, with the $x$ in each term.