MHB Coefficent in a infinite power series

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SUMMARY

The coefficient of \(x^9\) in the given infinite power series is definitively 10. This conclusion is drawn from the combinatorial analysis of the equation \(9 = 0 + 9, 1 + 8, 2 + 7, 3 + 6, 4 + 5, 5 + 4, 6 + 3, 7 + 2, 8 + 1, 9 + 0\), which identifies 10 distinct combinations that yield the term \(x^9\). Each combination represents a unique way to sum to 9 using non-negative integers, confirming the coefficient's value.

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  • Understanding of power series and their coefficients
  • Basic knowledge of combinatorial mathematics
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How do I find a coefficient of x^9 in a power series like this:
Screenshot 2021-06-03 174217.png
 
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For (a) 9= 0+ 9= 1+8= 2+ 7= 3+ 6= 4+ 5= 5+ 4= 6+ 3= 7+ 2= 8+ 1= 9+ 0.
There are 10 products that give $x^9$ so the coefficient is 10.
 

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