Combinatorics: solving for coefficient of x^n term

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SUMMARY

The discussion focuses on finding the coefficient of the x^10 term in the polynomial equation f(x) = (x + x^2 + x^3 + x^4 + x^5 + x^6)^3. The correct approach involves using the binomial theorem and manipulating the function into the form f(x) = x^3 * ((1 - x^6) / (1 - x))^3. The solution confirms that the coefficient is 27, aligning with previous calculations. The participant initially struggled with the binomial theorem but received clarification that significantly aided their understanding.

PREREQUISITES
  • Understanding of Discrete Mathematics concepts
  • Familiarity with the Binomial Theorem
  • Knowledge of generating functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the Binomial Theorem in detail
  • Explore generating functions and their applications in combinatorics
  • Practice problems involving polynomial expansions
  • Learn about combinatorial identities and their proofs
USEFUL FOR

This discussion is beneficial for students of Discrete Mathematics, particularly those studying combinatorics, as well as educators seeking to clarify polynomial coefficient extraction methods.

Armbru35
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Hi, I'm currently taking a Discrete Mathematics class and cannot seem to work out this one problem, we need to find the x^10 term in order to determine its coefficient of the equation f(x)=(x+x^2+x^3+x^4+x^5+x^6)^3 I know the answer is to be 27 from a previous problem (we are to use this method to verify our answer) but I can't seem to figure it out. I started with thinking of trying to solve x^3/((1-x)^3)-the sum of x^n starting with n≥7, but that doesn't seem to be working. Any suggestions would be appreciated!
 
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Hint:

[tex]f(x) = x^3 \left( \frac{1-x^6}{1-x} \right) ^ 3[/tex]
 
Ahhh...I was confused for a second but I was doing the binomial theorem wrong. Thank you so much that helped tremendously!
 

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